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The Big Bang is widely misunderstood in part because the phrase "Big Bang" was coined by someone who opposed the theory. It wasn't an explosion of matter, it is an ongoing expansion of space. Galaxies aren't moving apart because they are speeding through space away from each other, instead space is expanding between them. A common analogy that is used is to imagine that galaxies are dots on the surface of an expanding balloon. As the balloon expands, the dots get farther apart. The dots themselves are not moving across the surface. Now add one more dimension and that's pretty much what is happening.

There's quite a bit of hard evidence that the universe is in fact expanding. That tells us the universe was smaller in the past, and must have had a beginning, when it started out very small. There's hard evidence for that too.

"How big was the universe when the big bang occurred?"

According to the theory, the entire visible universe was once smaller than an atom. But we don't know how big the actual universe is beyond the visible horizon, so there's no way of telling.

But again, if the univese was once small (whatever that means), then the inflation + the speed of the galaxy through space must be near the speed of light, for earth to be in the position to receive light from 13.0 billion years ago.