The idea of space colonization is simply that life will continue to do what it's done since the beginning:
expand into every available ecological niche. Now that
technology has opened access to the cosmos, life from Earth will colonize space.
Why colonize space? First of all, why not? It's there, nobody's done it, and eventually it will be cheap enough for the restless to pursue just because they want to.
But there are good practical reasons too. We are so used to squabbling over the limited resources here on Earth that we entirely overlook the fact that our solar system is full of usable metal ores, organic compounds, and oxygen. A huge fusion power plant (the Sun) pumps out more energy than we can use. We are certainly doing a good job making more from less here on the home planet, but the developing world is going to want refrigerators and cars too.
Space seems far away, but in space, energy is what makes something far or near, not distance. That's because in the vacuum of space, something set moving will keep moving indefinitely--there's no friction to slow it down. Low Earth orbit, just outside the atmosphere, is only 120 miles away from where you sit now. But in energy terms, it's halfway to anywhere in the Universe. The energy required to reach low Earth orbit is half the energy required to reach escape velocity from the solar system. So once we get over that hump, reaching the Moon or the asteroids is only a matter of time. It doesn't require much more energy.
The asteroids are particularly good bet. They have no appreciable gravity of their own, unlike the Moon, so you can dock with an asteroid, mine it, and leave with little energy cost. Most of them are mountain-sized hunks of pure iron and nickel. The Moon is an especially good source of oxygen, locked up in the rocks. There might be hydrogen at the Lunar south pole too. Hydrogen and oxygen are needed to make water, but also for rocket fuel.
With all the environmental problems we have caused here on Earth with industry, doesn't it make sense to move manufacturing into space, outside the biosphere?
We are also running out of good places to live here on Earth. In space one builds real estate. The original space colony idea was developed by American physicist Gerard K. O'Neill with his students at Princeton in the 1970s. They realized that hollow spheres and tubes made much more efficient use of materials to house people than the solid spherical rock we live on now. The materials available on the Moon and the asteroids could be used to construct space habitats for an unlimited number of people. Until O'Neill's work, space colonization was usually understood as the colonization of planetary surfaces.
It's the ultimate libertarian dream come true as well. Just as the New World was colonized by dissident groups fleeing European oppression, every offbeat society can have its place in space.
The immediate obstacle to space colonization is the high cost of space transport. The necessary cost revolution is nowhere in sight. Perhaps advances in nanotechnology and automated manufacturing will enable economy of scale sometime in the 21st century. Perhaps not. The infrastructure for space colonization will also require a huge long-term investment. Perhaps the current global economic boom will lead to the resources and the audacity for such a project. Perhaps not.
And why would anyone want to live in an orbiting space can? The first thing to realize is that space habitats would eventually be kilometers across, city-sized, not the cramped quarters of today's space stations. They will be big enough to have weather. Of course, the prime real estate will still be Earth, the garden planet.