Why do we need deep-space travel? Not that I'm against it, but why would we need to go anywhere?
1) overpopulation
2) old
Sol is going to burn out in another couple billion years
3) Why not? We found and settled the New World, and explored the Polar regions. What else is there to do between crops?
1. We'd never be able to export population off the planet fast enough to solve overpopulation that way. Plus even if we could, there's the issues of who would go, and who stays. (Easier, I suppose if Earth were so miserable the decision to leave was a no-brainer.) However, space resources (Asteroid metals, Geostationary solar power, moving "dirty" industries off-planet, etc.) could provide a first-world standard of living to everyone on Earth, which seems to be the only proven way to stem population growth we've seen to date.
2. If we should survive until the sun leaves the main sequence, and there is some continuity of humanity that still survives until that time, we'll probably either possess the engineering prowess to deal with it, or we'll be so far evolved that such things won't harm us. One could argue a cyclical civilization, and "falls" or dark ages, but if that happens humanity will eventually succumb to climate change, disease, supervolcanisim, asteroid/comet impact, technological "oopses", or war long before then, well within a million years, or less. So if we survive until the sun itself starts changing, humanity (or what passes for it by then) by default will be able to deal with it.
3. Very true, some of the best success stories in our history are when we were challenged with "somewhere to go" and something to do. And unlike the Earth, we're much less likely to repeat the tragedies that accompanied the era of exploration. (i.e. "native populations") Space is huge, other sentient species, even if common, are still widely separated in both space and time. There is plenty of "elsewhere" to go to. And assuming we've mastered robust at-will interstellar travel, from a technological standpoint there is very little we couldn't produce ourselves or synthesize after only minor sampling and study. So even if our morality did not evolve, simple economics makes the Universe a very unlikely place for either conflict or exploitive colonization.
I disagree with some of the details, but in the general thrust of his arguments, Tallpine is right. If we don't expand off of Earth, humanity will go extinct. It's possible we could develop sufficient technology AND social organization to live on Earth indefinitely in peace and safety, but the historical record on the human scale does not make me very confident, and neither does the natural disaster record on the geological scale.
We are not "special". Forces beyond our control could remove humanity in an eye-blink. The mitochondrial DNA record of humanity's maternal line already indicates that humanity has already faced extinction several times, we've been winnowed down to a mere 20 or 30 fertile females in the past more than once.
As long as we are on Earth, and only Earth, all of our eggs metaphorically, and literally, are in one basket.