You've heard a slightly garbled account of the physics inside a black hole event horizon, but what you've heard is not so far from the truth.
The physics of (stationary) black holes is described by the Schwarzschild metric, though unless you're a GR nerd you'll find this a bit opaque. I'll try to describe what is going on in everyday terms, but you need to bear in mind that this is just an analogy and you can't get a full understanding unless you're prepared to learn the necessary maths.
To take your second question first, you're correct that at the event horizon the escape velocity is the speed of light. See this question for more info on this, and this somewhat more involved question for exactly why nothing can escape from the event horizon. Once you move inside the event horizon even light cannot move fast enough to avoid being dragged into the singularity at the centre of the black hole, and because nothing can move faster than light that means anything and everything that passes through the event horizon must end up at the centre of the black hole. That's why we get a point of infinite density. The density can only be finite if whatever has fallen through the event horizon is a non-zero distance away from the centre. But nothing can stay a finite distance away from the centre, so everything must end up exactly at the centre i.e. we have an infinite density.
Now to return to your first question, i.e. whether spacetime ceases to exist inside the black hole, spacetime does not cease to exist inside the event horizon but there is a sense in which it ceases to exist at the singularity at the centre. If you have some object travelling through spacetime, e.g. an apple falling towards Newton's head, then we can use simple physics to predict how the apple moves. That is, if we know where it is at time $t$ we can tell where it will be at a later time $t + \delta t$ for some small time increment $\delta t$. However if the apple is falling into a black hole we can calculate its trajectory right up to the point where it reaches the singularity at the centre of the black hole, but it is impossible to calculate its motion in space or time beyond this point. It is as though both space and time stop at the singularity. This is probably why you've heard that spacetime ceases to exist there. The technical term is geodesic incompleteness: the geodesic is just the trajectory in spacetime traced out by the falling apple.
Incidentally, few of us really believe the density becomes infinite at the singularity and that the geodesic of the apple ends there. Most of us think that some form of quantum gravity will blur out the singularity and keep everything finite. However, at the moment no-one knows what this theory of quantum gravity will turn out to be.