Evidence for the Big Bang
The BBT is not about the origin of the universe. Rather, its primary focus is the development of the universe over time.
BBT does not imply that the universe was ever point-like.
The origin of the universe was not an explosion of matter into already existing space.
Your first mistake was thinking that the BBT is an origin theory.
From quantum field theory, we know that something does indeed come from nothing: to wit, "vacuum fluctuations". In the simplest case, an electron, a positron and a photon can appear effectively out of nowhere, exist for a brief time and then annihilate, leaving no net creation of mass or energy. Experimental support for this sort of effect has been found from a number of different experiments. See, for instance, the Wikipedia page for the Casimir effect.
I can go on copying extracts from that source, but I'm sure you can read it for yourself.