This question really has two parts. First, how was matter able to get out of the big-bang singularity? After all, physicists describe a black hole singularity as a pit into which material flows but from which it cannot escape. Let us leave aside the fact that singularities are an idealization. The basic point is that the universe was born with a tendency to expand, which overcame the tendency of matter to collapse. According to relativity theory, space does not like to remain static; for all but the most special cases, it either expands or contracts. But why it initially chose the former is still a mystery. In some ways, you can think of the universe as a black hole turned inside out. A black hole is a singularity into which material flows. The universe is a singularity out of which material has flowed. A black hole is surrounded by an event horizon, a surface inside which we cannot see. The universe is surrounded by a cosmological horizon, a surface outside of which we cannot see. (A crucial difference, though, is that the event horizon is fixed whereas the cosmological horizon varies from observer to observer.) The second part of the question is: Why didnt matter in the early universe collapse into black holes? After all, physicists say that if you squeeze matter to a high enough density, it will collapse into a black hole, and the density of matter in the early universe was extremely high. The answer is that black-hole formation actually depends on the variation in density from one place to another--and there was very little variation back then. Matter was spread out almost perfectly smoothly. In fact, cosmologists usually turn the question around. The fact that the universe did not recollapse into a swarm of black holes is evidence that sharp density variations did not exist (or were extremely rare). This lack of sharp variations, in turn, is evidence for the inflationary model that most cosmologists today accept.
I know this is a lot but it will help you I promise that. I had to answer this same question and this is what I put.
