We know nothing at all about why things in general exist.
We know enough about the universe to rule out the idea that there's a big empty space, with matter rushing out to fill it. On a large scale, the space that exists has a fairly uniform density of matter. So there's no "vacuum cleaner" effect going on.
The question of what was before the Big Bang is being actively investigated. One type of idea involves a periodic colliding set of universes. If that is correct, some evidence should show up in features of the Cosmic Microwave Background, currently being measured.
The Big Bang did not result from matter, in the ordinary sense of the word. Instead, matter is a leftover from the Big Bang. We don't understand well why an imbalance between matter and antimatter developed after the Big Bang.
We don't know much about the ultimate origins of space. We do know that now it obeys the laws of General Relativity. Those say that if there's a background energy in space (whose origin is also not well understood) then the space expands faster and faster, if you use distance units in which atoms, etc. have fixed sizes.
(published on 08/09/11)