The very fact that there was a singularity, a point in spacetime where there was nothing, then there was something
That's not a fact though, it isn't even a theory, it's a plain misunderstanding of a vaguely educational misrepresentation of the wrong end of some good solid math.
Einstein gave us General Relativity, that's math that helps us describe observed phenomena like acceleration and gravitation in terms of the expansion and contraction of space. Hubbard provided doppler data of impossible proportions that stongly implied that space in general, the observable universe, was expanding rather than contracting, at a steady rate.
The next obvious step was to work that rate backwards and see what it could tell us about the early universe. Let's skip that sideshow for a second though and just keep working it back and see where we would have to stop. Turns out, duh, we have to stop just a smidgen short of zero volume for our mass, or else we end up with "infinite" density. Density being mass divided by volume, and "infinite" being x/0.
That doesn't mean that's where it starts though! It just means we can't start talking sense about it until one instant thereafterward, when it's just
near-infinite density. Get it? The observable universe isn't the whole shebang, the main point is that you can take the observable universe and unknown quantities further and stick it in a teacup, in a teaspoon, in a teeny tiny shape of an atom, and still have unknowable amounts more big shebang left for it to do its trick in.
But for convenience sake maybe we actually should stop short of infinite density? Hawking says No, infinity density probably actually happens a lot in a weird chaotic universe like ours. And assuming it does, these are the effects we would see. And black holes are quickly observed by means of the predicted phenomena. Black holes have infinite density, their mass has not even one smidgen between it and itself. And as a result they collapse spacetime nearby, and this isn't any sort of amazing philosophical headgame, actual infinities and zeros can be measured indirectly, they just can't be experienced by us because they collapse spacetime.
This not only didn't help, it hurt a little. Obviously we now definitely have to stop short of the infinite density, because we can't imagine getting out of one of those. The whole point with singularities is their inescapability. Like Hawking said for the longest time, nothing gets out of a black hole. So basically we couldn't have shot out of the black hole entirely, we would have to have been something like the Hawking radiation that we see escaping from matter as it gets sucked into the hole.
And yeah, that would be a whole lot of infinite density to make such a large explosion as to be the universe, it's quite out of scale with what we actually observe from the holes we can see. A whole shebang of its own would have had to collapsed into itself, to produce enough stuff escaping just before the horizon to be a little observable universe like ours is. Hence the whole Big Bang-Big Crunch scenario, you've actually covered that part yourself
But lately Hawking changed his mind. Lost a bet doing so, too, so it's a pretty sure thing. Now he says yeah stuff could someday come out, just nothing like the stuff that ever went in.
Doesn't really matter though. Never really was about the beginning, it was always about what the exercise could tell us about the early universe. Turns out one of the things that it does definitely tell us is that, at some point not quite that infinite density x/0 long ago, with all the mass-energy so close together but engaged in flying apart, every single quantum equation we can imagine ought to have been firing off all at once in the same locality. One of those possible fields, now dubbed the "inflaton" happens to be scalar, and so it supercedes all the others that are just additive or geometric or exponential without any trouble and expands space quietly and near-timelessly to a manageable scale where we can resort back to GR to figure things out.
Ergo, there was no "early universe", problem solved.
Anyway, notice how something never comes from nothing in this story? Finite measurable observable stuff comes from some unknowable larger background. Later, we improve our knowledge. We know more about the unknowable every day.
We used to not be able to do math with our infinities at all, just had to stop there. Cantor made math for us, it works great. In conjunction with observable quantum phenomena it clearly defines the physics that makes probability work. Weird things happen for a reason, infinities are dividing together all the time and producing quasi-unpredictable but easily mapped finite results. When you flip a coin, when you roll the dice, infinity / zero is happening right in front of you.