Well, most people now accept the idea of a big bang - an initial 'explosion' of matter, time and space from a single point, what scientists call a singularity. This explosion created everything in the Universe - including the space between matter, and time itself.
Expansion is the natural result of any explosion as particles are thrown away from each other.
In a sense, this is what's happening to the Universe except that it's not really the matter that's expanding, it's the space.