Many atheists would say that the Big Bang likely occurred, but taking this position is not a requirement of atheism. Whether or not someone is an atheist depends only on whether or not they believe that one or more gods exist – it doesn’t depend on anything else.
The Big Bang is a scientific theory that, in the distant past, the universe was much denser and much hotter than it is today. There are a number of pieces of evidence that support this theory. Among these are, firstly, that the universe appears to be undergoing a kind of expansion (as evidenced by the redshift of light from all distant galaxies), suggesting that in the past it was much more condensed, and, secondly, the Cosmic Microwave Background (a kind of afterglow from when the universe was much hotter, which is visible in all directions from Earth).
The theory of the Big Bang often also includes the idea that the universe was once contained in an infinitesimal point – of infinite density – called a singularity. However, there can be models of the Big Bang that do not include this aspect. (The physics of singularities in general, which includes black holes, is not well understood, so including this aspect is more speculative.) The ‘Big Bang’ refers only to the rapid outward expansion of the universe – not its possible origin as a singularity.
Many atheists consider the theory of the Big Bang to be a reasonable explanation of the physical evidence we observe in the universe. This is not the same as belief, however. If you believe something, you are convinced that it is true – you are certain. No reasonable atheist or scientist would claim to be certain that the Big Bang occurred – only that it is a good explanation of what we observe.
There are other ideas about the history of the universe – such as the ‘Big Bounce’ – which have some proponents, and an atheist might conclude that one of those is more likely true.
Note that the theory of the Big Bang is never a claim that the universe came from nothing, as is often suggested by opponents of atheism. It is simply the idea that the existing universe was once much, much denser than it is now.