The astrophysical observations of the motions galaxies, clusters of galaxies and of the microwave cosmic radiation led to the hypothesis that the Universe started at some point and has been expanding since then. This hypothesis has not been falsified by new data and has been going with the name Big Bang theory since the 1920s, from the kinematic analogy with an explosion.
These observations depend entirely on gravity, not elementary particles and their annihilation, and gravitation is described at large distances by General Relativity with Newtonian mechanics being the limiting case as distances get small. In particular the Big Bang has been modeled mathematically with in the framework of General Relativity .
In special relativity which is necessary to describe the behavior of elementary particles, it is energy that can be manifested as mass or radiation or kinetic energy. Thus given an amount energy there can be creation and annihilation of particles. This is in the micro domain. In the Big Bang model the expansion starts because the spacetime is changing, expanding.
In the first moments of the Big Bang the energy cannot be described by particles we know,only as the expansion cools the universe we can get the quark gluon plasma, which currently is being studied at the LHC. Thus particle creation is an effect of the expansion, not the cause of the expansion .
As the expansion continues the universe cools enough for the plasma to coagulate into protons and neutrons and electrons and their antiparticles. It is still a matter of research why there are particles only in our known universe.
These matters are described in the wiki article on the big bang, but to really understand them time should be spent in learning the mathematics necessary to understand the theories behind the model