> One point that was made in the video was that inside a black hole, space is collapsing faster than light. This seems like more than a change to coordinate system?

No, this refers to the fact that inside a black hole you can travel as fast as you want and do whatever you want but even if you move around at the speed of light, you still won't be able to escape the singularity. This is because the singularity is a so-called timelike singularity – it lies in your future. It's not a point in space somewhere that you can walk around and poke; while you're in the black hole, it's actually completely invisible to you until you hit it. You can basically compare the singularity to your own death: While you can't see it right here and right now but you know for sure that you'll reach it at some point in your future.

A physicsy way of saying all this is that, inside the black hole, all future-directed worldlines emanating from a given spacelike surface will converge and eventually hit the singularity. (This is what Nima Arkani-Hamed means when he says that space "collapses".) Another way of saying this is that the aforementioned surface is a so-called trapped surface.

Back to a more intuitive explanation: In a way, falling into a black hole and towards the singularity is like falling down a waterfall: Once you've stepped over the edge (crossed the event horizon), you certainly won't be able to stop yourself from falling anymore. You cannot fall up and escape gravity.

Here's a website which explains this way of looking at black holes as "waterfalls" of space a bit further: https://jila.colorado.edu/~ajsh/insidebh/waterfall.html

> Doesn't this mean that time is also moving faster than light?

As mentioned in my other comment, I recommend steering clear of the terms "time" and "space" as those are not individually well-defined and equally agreed upon by all observers. Instead, time and space are relative – if I use a different coordinate system than you, we will have very different notions of which events occur at the same point in time (and are thus part of the same spatial slice). It's called relativity for a reason. :)

So, my counter question to your question would be:

> Time as measured by whom? ;)

But even if you define time with respect to a given observer, it still does not "move" or have a "velocity" that you could compare to the speed of light. Only massive objects actually move through spacetime and only when they're close enough, you can compare their relative velocities.