In this case, you have to think of "expanding" as being like stretching. I.e., things within the infinitely big universe rapidly got farther apart from each other.
As I said above, at the time of the Big Bang, the universe was like a very dense sheet of rubber, and then it started rapidly stretching, getting less and less dense over time, until it is now the density that we see around us.
Infinities can definitely give you a headache, though. For instance, there are just as many odd integers as there are integers.
How's that? Won't they see a completely different universe due to the speed of light? If they're 14.6 billion light years away from us, and they in our direction, won't they see what was happening in this area 14.6 billion years ago?
If it is limited in size at all, it's because it wraps around on itself ("if you go straight long enough you end up where you were"). Nobody has firm evidence that it actually does wrap around on itself, and there's reason to believe that if it does, it's on a scale that's much larger than we could ever hope to observe. We will likely never have any sign inconsistent with it being literally infinite regardless of whether or not it is.
If you want a better answer than that, you'll need a very firm grasp of special relativity and a basic understanding of general relativity and the metric expansion of spacetime. In which case, read this: http://arxiv.org/pdf/astro-ph/0310808.pdf