My understanding:
Start with the concept that locations in space are separated not just by a measure of distance but also a length of time, the time needed for a phenomenon to propagate from one location to another. So if an object one light year away changes color from blue to red, then I can't know about it for one year, because that signal takes a year to reach me, and it is impossible for me to know about it before that time. Thus, I am not just one light-year away from it in space, but also one year away from the object in time.
So let's say I have a hypothetical infinitely long measuring tape, and I were to try and measure the distance between where I'm sitting now and the edge of the universe. What I'd see from my own position is that, as the other end of the tape gets closer and closer to the edge, it begins to compress so that the marks on the tape become closer and closer together, until eventually the distance between two marks is infinitesimal, that is, infinitely small. This represents space becoming smaller and smaller as I go further away from myself in distance, and therefore am measuring space as it was further back in time. The term "singularity" in this sense refers to this asymptotic decay: it couldn't have been 0, but it was infinitely small, but infinitely small is not 0. In theory, we can say that there was a "big bang" that happened about 13.7 billion years ago because that's when there is an asymptote in those calculations, ie, a "singularity" at -13.7 billion years. And so far, experiment agrees with theory.
And the reason we can say it happened everywhere at once is that the above is true no matter where I try to measure from.