Because we're human & it's there

We started off in the middle of the desert, yet we kept looking at the horizon and wondering what's there so some of us risked their life & limbs (which at that point really was all they had) & walked there to find out.

At some point we reached the ocean, and we got stuck for a while, that is until some clever cave man figured out that a tree that fall to the water in a recent storm floated, he stood at it and it still floated, suddenly the ocean, a mass body of water that no man can live in longer then he can swim (IE not long) and where every failure was deadly was possible, he didn't know what was behind the ocean (or even if there is something beyond the ocean) but that didn't stopped some of them to take the risk and get on fallen trees and try they luck.

Time went by and the trees became large ships, crossing the ocean wasn't a big deal anymore so we started to look below it, sticking ourselves in what is essentially metal coffins just to go and look what the bottom of it looks like.

But even that wasn't enough for us, birds can fly, we wondered what that feels like so we created planes to let us fly alongside them, it may have other uses now but at first it was just to fly for flying sake.

Then we reached space, the final frontier, and like every frontier before it we are now there as well... not far away yet but our wanderlust knows no bounds, we have footsteps on the Moon & robots on Mars, we are actively working (and spending billions) all around the world (and working together, just look at the international space station) to colonize both in the not so distant future (sure most plans are still in the design stage but even the wright brothers started out that way).

I promise you that once we get to a point we can have a human being live full time on another part of the solar system (which given your question states there's no need to worry about the tech needed I assume it exists in your world), you will still have people looking to other stars and saying "I want to go there", I know that because I am one of those people.

Reason? we don't need a reason, it's there and we're not and that's all the reason we need.

The survival of our species

What happens if we suddenly have a nuclear war and all of us die? Or if the sun explodes? Our species would become extinct. That is why we need to colonize at least a second planet in another star. By that way, we will exponentially increase the survival chances of our species from mass extinction.

War. Survival from other lifeforms

We aren't alone in the universe. It's very likely that there is something there, watching us or taking the galaxy. Just because we can't see any other lifeform it doesn't mean that the Fermi Paradox is true. Maybe all of them are hiding. If there exist several lifeforms in the whole universe, it's obvious that at some point we will engage a war with them for the resources of the galaxy and possibly the whole universe.
We should no longer rest, instead, we must start building massive spaceships to colonize as quick as possible the whole galaxy before others do that! Prepare the ships!

One of the things oft ignored in science fiction, is that once such a ship leaves on its journey, the cost of doing so becomes nil.

That is, the economy on the ship is entirely self-contained. Talking about 'earth dollars', 'earth salary', and 'earth living costs' is meaningless. There ARE no salaries paid from earth. There ARE no support costs from earth. The inhabitants of the spaceship are as rich as they ever will be, in earth dollars. That is, absolutely penniless. But in terms of PPP, they would be as rich as the collective manufacturing capability of the spaceship. They could never own more than can be produced (or originally supplied) on the ship.

The fact that they would never be PAID again is, of course, moot, because there is nothing for them to SPEND their income on. They can't send it back to earth.

So one benefit, and a not insignificant one, is that these inhabitants, and their descendants, would be living 'free' in any earth-based sense. They, and their descendents, would never have to pay rent, a mortgage, food, car loans, or any other expense ever again.

The potential to live in a perfect socialist commune type community, forever free from the yoke of money and of conceding it to others, is a very powerful motivator for many.

In a sense. it would be the perfect 'convent', the perfect 'cluster'. And it pretty much guarantees that all of your descendants would live by the same philosophy.

TL:DR

It would be very appealing to a distinct ideological group who wished to enshrine their belief system in an environment that ensured it could not be altered, immune from any social or economic pressures from any other ideological factions on earth.

Incidentally, there are a good selection of such ideological groupings on earth that are sufficiently wealthy enough to pull off such a feat.

In the near future, probably nothing. In medium term future, there is a number of possibilities.

Today, building a starship like that is simply impossible. But in the near future, when we'd be able to sort out all technical hurdles of sending humans to Mars, a giant interstellar spaceship will become a theoretical possibility. However, there would still be a number of issues that would make a success of such project highly unlikely.

1. Propulsion. Existing (or practically achievable) methods are too
   slow for interstellar trip, likely making it to last tens of
   thousand years. Even if everything else is done perfectly, we just
   can't expect our generational ship to function that long without a
   failure;


2. Energy. Solar panels won't be effective in interstellar space, and nuclear fission reactors are the only currently available option. This ship would have energy needs much much higher than a Voyager-type probe, which might not be achievable with current tech level;


3. Autonomy. 10,000+ years level of autonomy is impossible to get at our present tech level. 1,000 years maybe is realistic, but still something that we can only hope for;


4. Destination research. Right now we have a very limited ability to detect Earth-like planets in a target systems, and no way of telling if such a planet is a good candidate for colonization;


5. Money. As you have mentioned, this would be a huge undertaking, much bigger than the Apollo project, for example.

So, I would think that even under the best conditions, building such ship in the next 50+ years is impossible. Of course we can try, but the project would have a very low chance of success. I can not think of any realistic calamity that would affect entire Solar system and give us enough notice to build this spaceship.

However, looking further forward, all of the issues in the list above can be eliminated or somehow mitigated. New types of engines can be built, thermonuclear energy become reality, deep space experience would turn into confidence in spaceship's long term durability, our telescopes will find very Earthlike planets and new technologies like space elevator would make building spaceships in orbit very cheap.

If the complexity of this project is reduced from the "Tower of Babel" to "Apollo" level, then we can find a number of possible reasons to complete it - from below-extinction level catastrophes to simple curiosity.

I'd like to suggest a minor frame challenge.

While it is unlikely to the point of impossibility that nations would sacrifice a major portion of their GDP to send a millennia long mission to another star, and a world-ending catastrophe is likely to be too fast and too destabilising to allow the construction of such a vessel, there are alternatives.

1. The Eccentric Trillionaire

In the relatively near future, space mining will likely become a booming industry, and the world will see the rise of trillionaire space executives. It is entirely feasible that such a person could decide that they'd like to invest their trillions in constructing an interstellar starship, and they'd certainly have the resources to do it.

2. The Religious Exodus

Another kind of organisation that could decide to depart the solar system without care for the cost nor ROI, is a religious order with enough funds to achieve it. Think something like Scientology, or the Church of Later Day Saints. They don't necessarily need a concrete reason, they just need their followers to believe that it is the right thing to do.

Any unavoidable world-ending event, such as a massive meteor hurtling towards earth with a 0% chance of survival. In such an event, the 500 people would be the highest tier professionals in the world in their respective fields, and most likely be people who also designed and built this massive ship, in case of repairs needed. In my head, it has to be something catastrophic to human life on Earth. Nothing less would cause all countries to put money towards any single thing.

If you watch the read the expanse series by James A Corey (or watch the TV series if you don't read great books!) then you come across another reason. The Church of Jesus Christ of latter-day Saints are building a generation ship to travel to another system.

The freedom to take your religion and go to a fresh new world and live under your own rules free from the rest of the world would be extremely attractive to most religions, cults and sub-cultures. The only issue is who can afford it. I would suggest that organised religions are actually the most likely to be able to afford this (the catholic church owns a huge and incredibly wealthy land portfolio that is larger than the GDP of several countries).

The more a minority is "oppressed" by the majority the higher their motivation to flee to somewhere they can live as they want. The only thing stopping all sorts of groups going off to try and set up their own ideal society is the economics. As noted above in other answers there are the curious and scientists also who would go just to see what was there and expand our knowledge of the universe.

Leaving earth forever and having generations live and die en route is a huge price to pay but millions of people would choose to pay that price and commit their descendants to paying it also. Also don't forget how many people have pretty crappy poverty driven lives here on earth. A lifetime (relatively) safe constrained in a spaceship with food, books, tv and relative comfort is better than what a lot of people currently survive with on earth. Plenty of volunteers if you can solve the economic problems.

This was the "economic incentive policy" that was developed after the "YYYY World Summit".

It has been deemed "coincidental" that this project was inseminated during the same summit that saw the legalization of Marihuana world wide.

Good night Sun

In the future, the sun is not going to keep being the life sustaining ball of light as we know it to be.

1. First, it will increase temperature, becoming brighter and brighter. This will make Earth life as we currently know it impossible.


2. Next, it will grow into a red giant, consuming the Earth. Earth life in any form is know impossible. That also means all the resources on the Earth (including all known organic materials) will be lost.


3. Finally, the sun becomes and white and then a black dwarf. Now, life in the solar system has a time limit, since there is no longer a source of energy.

So, at some point along this path we will have to flea the solar system or die. Humans do not like to die usually, so we will probably choose the former. Luckily, it should be feasible by then.

Now, as for whether we will choose the heat death or something else? THERE IS AS YET INSUFFICIENT DATA FOR A MEANINGFUL ANSWER.

tl;dr- The only real reasons for trying to establish a distant colony today would follow from long-term interests. People who care about such long-term interests could be directly motivated, while others could be indirectly motivated.

Direct motivations: Reasons for those who care about the long-term

Reasons for humanity to spread out across the stars include:

1. Survival of species and our legacy.
   If something would destroy Earth or the solar system today, that'd pretty much be it for humanity. If there are human colonies elsewhere, then our descendants would live on.




2. Scientific exploration.
   There's a ton we don't know about the universe. Projects like establishing a trans-stellar civilization are likely to help chip away at our own ignorance.




3. Greater technological output.
   While it's unlikely that trading most material goods could be economical any time soon, our economy's increasingly interested in information anyway. Scientific/medical discoveries, technological advancement, and other high-value information could be generated and shared between planets.

Indirect motivations: Reasons for those who don't care about the long-term

The concerns are fundamentally about long-term interests like those above. Anyone who doesn't care about such long-term interests presumably wouldn't be interested in any action that won't see fruit any time soon following from their own motivations.

However, even those who don't care about such long-term interests may still be indirectly motivated to pursue them.

Examples:

1. Elon Musk claims to want to go to Mars to help ensure humanity's survival. This is a long-term interest that many don't share. However, as a billionaire who's established a company that can pay hefty salaries, Musk's own motivations can inspire others to pursue the long-term interests anyway, if only for a paycheck.




2. The United States won the space race, getting to the Moon first. Which, if you don't actually care about getting to the Moon or any implications of it, doesn't actually matter to you – directly. However, there are people who do care, motivating the initial concern, while others got swept up in the goal through indirect mechanisms.




3. Stocks and other investments. If you have a large amount of money, e.g. in a retirement fund, you likely have stocks and other investments in which your actual money has been given to others for those others to pursue goals you may not know about, including some which you may agree with and others which you may not. Either way, it doesn't seem to matter; you have indirect reasons for funding those ventures, e.g. you want money for retirement later, so you've ended up giving your hard-earned money away to those projects despite not having direct motivation to have done so.




4. Bitcoin, as a specific case of the above example. A lot of people seem to buy Bitcoin out of the belief that others will want it and pay for it later, despite not caring for the use cases or/and ideals behind cryptocurrency themselves. This is, a lot of people have ended up paying their hard-earned money to enter the Bitcoin project despite a lack of concern for its goals.

Physics analogy: Direct action causing indirect actions

Here's a figure of a man pulling a woman on a sled:
$hspace{150px}$
$hspace{450px}$$llap{-text{source}}$.
Above, why does much of the sled and the woman's body move? Surely they don't care about the action of the man nor the rope directly.

It's an indirect effect, where the motive force propagates through the system.

Likewise, the core motivation for starting a project to travel to a distant solar system is based in long-term interests; there's no other direct reason that makes sense. However, all sorts of indirect motives follow from that, ranging from stuff like getting employed to pursue those tasks to national/etc. pride in it, to simply getting caught up in the enthusiasm of such a voyage as entertainment.

Point being that the direct motivations are long-term, though indirect motivation can still pull the bulk of society even if the bulk of society has no direct concern for them.

Conclusion: Long-term direct motivations are the ultimate reasons.

Ultimately, the reasons to go to another star are the long-term stuff. Humanity's survival, scientific exploration, greater technological output, etc., are the only core reasons that survive in a vacuum.

However, a lot of people may end up supporting a distant space colony if only for indirect reasons such as

1. to be part of a social movement that's excited about it;




2. to be paid for working on it;




3. to show off as a mode of self-advertising;




4. to have fun doing something that others care about;




5. to have a stake that others might pay them for later as an investment.

If you're writing a story about something like a space colony project being launched in the near-future, it's likely that most people in that world will buy into it for an indirect reason rather than actually being concerned about the core motivations themselves.