See, this is why I took Astronomy instead of Biology or Chemistry for my required science classes in college. It's just so much more fascinating! Mind you, I only took two semesters, so the following is what I was taught - perhaps they kept the more complicated theories from us "newbies." If someone happens to know more than I do, please feel free to refute my answer.

Think of it this way. You have a balloon. You draw a few dots on the balloon about an inch from each other, and blow it up to max size. All of a sudden, each dot is now five or six inches away from each other, but technically nothing changed about the structure of the balloon. (Yes, there's air inside the balloon, and a few other things that get in the way of technically explaining this, but ignore those for a bit.) That's basically how space is expanding. The actual space itself is expanding - it's not like a galaxy decides to meander farther away, the actual space (including the galaxy) expands.

As far as we know, that's how it works. It is, I suppose, possible for space to be expanding into something (due to the speed of light, we can't see past a certain distance away, since the light just hasn't reached us yet), but then what is on the other side of that space? And on the other side of that one? You see, it's either finite or infinite - neither of which make any sense to us.

Oh that is such a good question... I would really like to know that myself...