Let's forget for the moment the geometry of universe as it brings us into the nature of time and just confuses the issue.
For the sake of simplicity let's do a few things:
- Consider the universe to be a really really big ball.
- Assume that before the Big Bang, there was nothing.
- Assume that spacetime is flat (it isn't, but again just confuses the issue).
At the time of the Big Bang, matter was ejected in all directions. However more than matter was ejected, but also light.
As you know, nothing can travel faster than light. Therefore, the light that was ejected at the time of the Big Bang defines the borders of the universe.
Imagine this: Nothing exists except a single lightbulb. There is nothing else. Turn that lightbulb on. Light will begin to emit from that light bulb in all directions.
Let that lightbulb go for one second then we'll stop time. At this point, the universe is exactly 2 light-seconds in diameter.
As we travel around our time-stopped 2 light-second universe we are actually looking at the universe at various points in its short 1 second life. As we get closer to the light bulb, we see the lightbulb as it was very very recently. As we move away from the lightbulb, we see things as they happened closer and closer to the point where we turned the light bulb on.
When we get 1 light-second away from our lightbulb we see it just after it was turned on. But if we move just a little further away, we can't see the lightbulb anymore. Light from the lightbulb hasn't gotten there yet and therefore we are looking at a time before time began in our little universe. We are now outside our universe.
Now back to our own reality. Since nothing can travel faster than light and light expended at the time of the big bang is the furthest away, the size of the universe is bordered by that light.
Update for additional comments:
I don't believe anyone really is saying the universe is "infinite" by saying it has no bounds. In fact, there scientific community isn't arguing about whether or not the universe is infinite, but rather how big it is finitely.
One could say that it is functionally infinite because even if you could break pretty much every law of physics, go faster than the speed of light, and somehow pass beyond the original photons from the Big Bang, you'll have moved into a space of where nothing exists (yet), including time itself.
See this link for a better explanation: http://en.wikipedia.org/wiki/Cosmic_inflation