|MadSci Network: Astronomy|
Good question! The short answer is that the universe didn't actually begin as a finite point. Unfortunately, this is not at all clear given the language that astronomers and cosmologists use to describe their models.
The chief culprit is the term "Big Bang" (which was actually made up by Fred Hoyle, an active opponent of the Big Bang model). The name implies that the universe began as a point which exploded outwards. In fact, this is not a part of the Big Bang description! The Big Bang model simply claims that the universe was hotter in the past, and that space is expanding. This expansion cools the universe and makes it less dense. Imagine an infinite plane (this plane corresponds to the universe at the beginning of time), and label two nearby points. Then imagine stretching the plane, so that the two points get farther apart. That is how the Big Bang would occur in an "open" or "flat" (i.e., infinite) universe. It is always infinite; expansion mens that points get farther apart as time passes. The Big Bang model itself describes how the universe cools as it expands, and what happens to matter and radiation as the expansion progresses.
On the other hand, the observable universe did begin at a point. This follows from projecting Hubble's law (which states that expansion speed is proportional to the distance from us) back in time. Far enough back in time, all the galaxies will come together. That's okay though, because, as you say, the observable universe does have a finite volume! The trick is that this is only a tiny part of the whole infinite space.
For more information, and a helpful diagram, see Ned Wright's Cosmology FAQ page .
Try the links in the MadSci Library for more information on Astronomy.