I could have written the Carl-Sagan-styled shit, like "the space is a final frontier, and the constant curiosity and research is the purpose of our existence". But I won't. Instead, let's just be pragmatic and rational.
We'll never be able to "live well" and peacefully get along with each other, regardless of whether we "conquer" other planets, invest billions in building a particle supercollider or a soccer-field-sized telescope.
In fact, we've been spending more resources on "not getting along with each other" (wars, arming other countries, training soldiers, etc), than we'll ever spend on science in an upcoming century. So these two things are not even closely correlating, since the investments are incomparable.
If we pay money to satisfy the blood-lust hobbies of a bunch of uneducated army-guys, that cannot realize themselves by any other means, then I don't see any reason why we shouldn't invest at least some money in the peaceful hobby (called science) of a little bit more educated geeky guys.
On the other hand, technologies that are boosted by the scientific progress, especially space technologies, are good investments into the new toys for army-guys. In other words, why shouldn't we conquer other planets and continue not living well there? So maybe even we should conquer other planets not despite, but because we are not able to be friendly with each other.
There is a common argument, that the collaboration in space might be helpful for diplomatic relationships between countries. I really doubt it. The future of space missions is probably going to be taken over by the international private (or mostly private) companies that have nothing to do with any political or religious agenda.