I think that's a good question. It is natural to require that the total charge in the universe was zero at the start of its life (same goes for all quantum numbers). Otherwise, there was something before there was anything! Seriously, either this matter-antimatter asymmetry was an initial condition (in which case we can't hope to understand it) or it appeared at some point during the course of things.

The problem of finding the exact mechanism that gave us the matter-filled universe we observe today, with all the antimatter gone is an open question. There are a bunch of hypotheses, and experiments coming son to a large hadron collider near you will help physicists gather pieces of the puzzle.

The big-bang did not create particles or atoms, merely energy. In the very initial stages of the big-bang all of the mass of the entire Universe was locked up in bosons (photons, W & Z bosons, etc.), with virtual quarks and other exotic particles existing only ephemerally. At this stage the Universe is essentially an enormously hot photon gas. As the Universe cooled lower energy particles (such as electrons and protons) became more stable, and more and more energy started to become locked up in them. Interactions involving high energy photons can create particle / anti-particle pairs, in theory these should be exactly balanced. However, in practice it appears as though there is much more matter than anti-matter in the Universe. Likely this is due to some slight imbalance in particle/anti-particle pair creation, but we do not sufficiently understand this (though there have been a few hints).