We don't know for sure, but it certainly fits into our theories. There is, of course, no way to actually test if the Universe is infinite, but right now we think it is likely.
Also, if you read my updated answer on your other post, the Universe has always been infinite in size. I explain over there how it actually works: space is created in between everything, and thus one could say the Universe is expanding.
These objects can actually drift away from each other faster than the speed of light. That is, light from them eventually won't make it to us, since they'll be drifting away too quickly.
Now, this doesn't actually go against Einstein's theory that the speed of light is the fastest thing in the Universe. Einstein said that nothing can travel through space faster than light — but here, space itself is actually being created between the objects. Distances are increasing because space itself is dilating, and thus we can drift apart from other objects faster than light.
Really, there is no limit (as far as I know) to how fast we can drift away. Farther objects will keep drifting faster and faster away, since our gravity has a much weaker effect on them.