Event horizons are everywhere
The top-rated answers both talk about "tipping light cones" as though that were a special phenomenon that happens in black holes. In reality, nothing special happens at an event horizon, and you see your feet there for the same reason you see your feet anywhere else. In fact, event horizons (one-way surfaces in spacetime) are literally everywhere. Your feet are falling through one right now.
Forget about black holes for a moment. In fact, forget about general relativity, and consider a 1+1 dimensional special-relativistic world, with a couple of astronauts in it (Alice and Bob), and a diagonal line drawn arbitrarily through it:
| /. ^
| /. |
| /. future
| / .
| / .
/| . past
/ | . |
/ | . v
/ | .
E A B
E is the worldline of a point (or, in 3+1 dimensions, a plane) moving rightward at the speed of light. Alice crosses that worldline (feet first—i.e., her feet are on the left), while Bob fires his thrusters and goes into hyperbolic motion which avoids crossing E indefinitely. (Again, E is not a physical object, just a line I've drawn, but Bob can fire his thrusters whenever he wants, and chooses to fire them here for whatever reason.)
E is an event horizon. It should be easy to see that once Alice has crossed it, she can never go back, even if she can travel at the speed of light. Likewise (and for the same reason), if Bob never crosses the horizon then he will never see any light (or anything else) from Alice after she crosses the horizon.
Most of the "phenomenology" of black hole event horizons applies also to this horizon. Bob will see Alice "frozen" on the horizon at the moment of falling through, redshifted into the indefinite future. (If you don't see why, just look at the path of the light.) Black hole horizons behave like electromagnetic conductors; so does this horizon. Black hole horizons emit Hawking radiation; so does this horizon (it's called Unruh radiation, but it's just the infinite-radius limit of Hawking radiation). For the most part, if you want to know what happens at a black hole horizon, you can work it out in this special-relativistic analogue. (The exception is when a large part of the area of the horizon is involved, so that the spherical shape is significant. E is the infinite-mass (infinite-radius) limit of a black hole horizon.)
Obviously, Alice sees her feet at all times. That's not because of tilting light cones or because she's moving very quickly. It's because her feet are radiating light at all times, and it's never blocked by anything. Just before (resp. after) her head crosses E, she sees her feet just before (resp. after) they cross E. Bob never sees the light from after she crosses E, but that's only because he's moving in such a way that it never reaches him, not because it was absorbed or otherwise blocked.
What makes black hole horizons special is the (generally unknowable) future
Black hole event horizons (unlike E) are not at arbitrary locations, but that isn't because anything happens at the location of the horizon. It's because there is a future singularity where everything is destroyed.
This also has a special-relativistic analogue. Imagine that a region of space is peppered with undisarmable time bombs with synchronized countdown timers. The spacetime diagram looks like this:
\ / |
\ / future
Each asterisk is an explosion. The diagonal lines bound the region of spacetime from which you can't avoid being blown up, even if you can travel at the speed of light. Those lines are meaningful—their location is fixed by a real physical event—but there is nothing there. Nothing detectable happens when you cross them. Nor is there anything to detect anywhere inside the certain-death region. The causal future (future light cone) of the explosions is entirely outside that region, so no one inside it can detect the explosions in any way. They might infer that they are going to happen if they can see the bombs and countdown timers (which is analogous to noticing the rapidly increasing tidal forces), but they will never see (or otherwise detect) their feet blowing up before their head does. Because the explosions are spacelike separated, there is no sense in which their feet do blow up before their head, even if they crossed the certain-death horizon feet first.