What Schwarzschild found was that this solution had two singularities. There was a singularity at the origin, and a singularity some distance away from the origin, at a radius which is proportional to the mass of the object. In both cases, some term of Einstein’s equations become infinite, and therefore seem to stop describing reality.
A couple of years later other physicists proved that by changing the coordinate system in a certain way, that second singularity would go away. That is, this singularity was merely an artifact of one of the simplifying assumptions that Schwarzschild started with, and didn’t represent any real feature of a real black hole. But the radius at which that singularity occurred is still relevant: it tells you how big the event horizon is.
Penrose and Hawking proved that the singularity at the center of the black hole cannot merely be a mathematical artifact. Penrose got his Nobel prize for that.
> Is the math referring to the center of gravity as the singularity? Because even a sub black hole mass object would have that.
Yes, the singularity at the center of a black hole is coincident with its center of mass. But not every center of mass is a singularity. Remember, we only have a singularity if some term of Einstein’s equations for General Relativity goes infinite. In a normal object nothing is infinite at the center of mass. That only starts to happen once the object is compressed into a black hole.
> Why is there an assumption that just after a mass becomes a black hole the matter inside it suddenly compresses further when the actual gravity of the object has only slightly increased?
There isn’t. This is the most common _conclusion_, based on the math. It’s not a starting assumption.
> Is there a maximum density of matter in the universe and if the black hole even reaches that?
Unknown. A neutron star is the densest object that can exist which doesn’t have an event horizon. Get any denser, and an event horizon forms which completely hides whatever happens next.
> Wouldn't you need that number to be infinite if the black hole itself is infinitely small?
Sure, if the mass all falls into the singularity, then the density of the singularity is infinite. There’s really no problem with that.
> If the black hole does have a mass inside it... Do the light particles trapped inside the black hole form a blanket around the existing matter?
This is a non–sequitur. It’s meaningless.
What is really going on here is that mass bends spacetime. This causes our path through spacetime to bend as well. All the gee–whiz effects of relativity, like time dilation, are a result of this bending. Time passes slower for an object moving quickly because its path has been bent so much that it is passing through less of the time dimension than it would otherwise. Light is moving so fast that it doesn’t pass through time at all; from the perspective of the photon no time at all passes between when it was emitted and when it was absorbed, no matter how far it traveled through the intervening space between those events.
Inside the event horizon of a black hole, the path of every object is bent towards the singularity. They’re bent so much that time and space swap roles. The singularity is inescapable not because something is dragging you towards it, but because it is literally in your future. From the outside it looks like the singularity is a place, but if you get close enough it becomes a time in your future, beyond which there is no more time. Everything outside the black hole is twisted around into the past, where you can never go. And, unfortunately for you, a stellar mass black hole is not very big. It’s only a few dozen miles to the center, and since we move through time at the speed of light, that is a very short timeline indeed. Your future ends very abruptly, probably faster than thought.
That’s why most people conclude that the matter is crushed into the singularity. How could it resist? You’re not going to brace yourself against the walls of the universe and prevent yourself from being dragged forward into the future, no matter how strong you are.
But, quantum gravity might change that picture slightly. It may be that the distance to the singularity grows the longer the black hole exists. It may in fact grow at the speed of light, meaning that while the volume of the black hole is not infinite, you never actually reach the singularity and go splat. Your time never actually runs out. You can find some lectures by Susskind about this on Youtube if you like.