Imagine a soap bubble. Imagine the two-dimensional space on the surface of that soap bubble. Now imagine the soap bubble is leaky, and the air is escaping from it, making it get smaller. Eventually that two-dimensional space will have zero size, and anything that lives on it can only do so if it also has zero size. Of course, the soap bubbles in our everyday experience live within a bigger, three-dimensional space, and any little bugs that might live on them could just jump off. The difference with the three-dimensional space of our universe is, as far as the data has told us do far, there isn't any four-dimensional space in which our three-dimensional space lives. Still, the three-dimensional space of our universe can in principle curl up to have zero size.

Another point, though: it isn't exactly true that the big bang singularity had zero size. I'm not saying it had nonzero size. The real issue is that people are naively led to the wrong question. The singularity is a feature in our universe where the very concept of geometry breaks down. You can't really say what the "size" is of something that doesn't have any geometry at all. A better question is: what does the universe look like as one traces back in time toward the singularity. The answer to that question is surprisingly interesting, but also surprisingly technical.