It seems like there’s sort of two separate questions here:
First, why is the universe not centered around the point where the Big Bang happened? This has been pretty well answered already: Basically, at the time of the big bang, the whole universe was just a single point (meaning space itself, not just the matter in space), so the Big Bang actually happened everywhere in the universe. We can’t look at our universe today and say “The Big Bang happened over there” or “The Big Bang happened over here,” because at the time of the Big Bang, “over there” and “over here” where the same place.
But this still leaves one to wonder why matter isn’t uniformly distributed throught the universe. If the Big Bang happened everywhere, one might think that there’d be just as much matter at every point in space. But instead, we have galaxies in some spots and big empty spaces in others. As I understand it, this is believed to be due to quantum fluctuations. In layman’s terms, quantum mechanics says that there are tons of random fluctuations happening all the time, but at such an incredibly tiny scale that human beings don’t notice them in our daily lives. But at some point shortly after the Big Bang, the universe was so small that these fluctuations were on a scale that they actually made a difference in the overall distribution of matter and energy in the universe. Then the universe went through a period of very rapid expansion, and all these tiny density variations were blown up to a scale where they could eventually lead to the formation of galaxies.