So does that mean Earth's event horizon right now is also 8mm from its center? Like, if the earth was made of (nonrefractive-, same-mass-as-the-earth-now-) glass, would we see a tiny black hole down in the center?

No, by crushing the earth you increase it's mass (density). As the mass increases the gravity surrounding it increases.

What they're saying is if you could compact the earth to that size, it would be so dense it's mass would be high enough to have enough gravity to create a blackhole.

One wouldn't exist because that entire mass of non-refractive glass would need to be within that little radius to form a black hole. Until that point an event horizon wouldn't exist.

Increasing mass and increasing density isn't the same (that's the point he makes in the video about earth not being swallowed if the sun became a black hole).
My point was that the Earth's center of gravity should have an event horizon, but yeah, now I realize that thinking "center of gravity" is like thinking "geometrical point in space with the mass of the Earth", i.e. a black hole. The glass thing wouldn't work, my bad.

Let's get one thing out of the way first: If the Earth should somehow be compressed enough to form a black hole, its mass would stay exactly the same. What makes black holes so spectacular is not their huge mass, but the fact that their mass is compressed into a very small area. This again means that you are able to get dangerously close to the center of gravity. This is not possible here at Earth, since there is a lot of rock and magma preventing you from getting that close.

But even if you were able to burrow yourself down to the center of the earth, you still wouldn't be swallowed by a black hole. This is because the amount of Earth dragging you downwards would be much less, and all of the rock above you would be dragging you upwards. You would in fact be weightless, since all the parts of the Earth would be dragging you in opposing directions. You would still be crushed because of the high pressure, but that is another story.

All massive objects have a property called the escape velocity, which is the speed you need to escape its gravitational well starting from its surface. And this property would increase if you compressed the Earth, since its surface would then come closer to its center of gravity. If you compressed it to a radius of 8mm, this velocity would excede the speed of light, which means that nothing, not even light, could escape.

This is the definition of a black hole.

Some of the animations you see on his screen can be seen in more detail here:

http://science.videosift.com/video/Black-Hole-at-the-Galactic-Core