Looking at this thread, I see that the OP has asked a question, 20 answers have been given, the OP hasn't posted again, and virtually all of the answers are not answers to the OP's question. The OP asked, "how do we know X?," and virtually all the answers are "X is true! X is true! Let me explain X to you!"
twofish-quant said:
Let me rephase that to something stronger. The observations say that for the parts of the universe we have any data of, that the universe is not spreading into a void. If the universe was spreading into a void, you'd see the effects of the void. You don't.
Twofish-quant has tried to answer the OP's actual question here. Hurrah!
I'm not completely satisfied with twofish-quant's answer, however. Maybe it could be improved upon.
Our observations can only reach out to a certain distance. If all we know is that we don't see a surrounding void, that doesn't seem like a very strong argument to me, since maybe we just don't see the surrounding void because it's too far away.
Here is a possible alternative approach.
(1) General relativity has passed a variety of experimental tests, so we think it's probably pretty accurate. (2) We observe that the redshifts of distant supernovae follow a certain dependence on their distance from us. (3) We also observe that the cosmic microwave background has fluctuations on certain angular scales.
Smart people have tried very hard to find a model that fits 1, 2, and 3. All they've managed to come up with is a particular model in which there is no preexisting void. In fact, the observations fit that model extremely well. On the other hand, nobody has ever found a model *with* a preexisting void that fits 1, 2, and 3. That makes us suspect that there is no preexisting void.
Another line of attack is that if we only assume 1 above, then the Hawking singularity theorem
http://en.wikipedia.org/wiki/Penrose–Hawking_singularity_theorems , plus some relatively crude observations of the present state of the universe, tells us there must have been a singularity in our past. Such a singularity, as described by GR, has features that are incompatible with the idea of an explosion in a preexisting spacetime. In particular, if GR is correct, then timelike world-lines can't be extended backward through that singularity.