Really what it means is that past the event horizon you can use the direction in space between you and the singularity as a way to measure time, specifically the amount of time left before you reach the singularity. It's not so mind blowing when you interpret it that way now is it? You can imagine many things in ordinary life that you use to measure time without claiming that time has literally swapped places with it. On a road trip, the number of kilometres to your exit tells you how long you have left, that's using space as a proxy for time... big deal. The notable difference between a road trip and a black hole is that on a road trip you could stop for a break, you could maybe take a detour, you could decide to go back home... and these would all break your use of space as a proxy for measuring time. Well with a blackhole you can't do any of those things, there is no going back, there is no detour, the relationship between the spatial direction towards the singularity and time is fixed and causal and there's nothing you can do about it.
The phrasing used is used almost certainly to evoke some kind of voodoo mind-blowing mystery that completely disappears when you get down to the more strict formalism.
That's not correct. There is a relationship between the radial coordinate r you are at and the time it will take you, by your clock, to reach the singularity (at least assuming you are freely falling), but that relationship can't be described the way you are describing it.
To put the issue with what you say as starkly as possible: at any event inside the black hole's horizon, there are spacelike curves in every direction from you that will hit the singularity! So "the direction in space between you and the singularity" is not even well-defined.
To be clear, you are right that Susskind's statement is very misleading (I posted my own criticism of it upthread).
Note that, once you're inside the horizon, you can't "turn around" and go back outside again. You're inside the hole for good.
And once you're inside the hole, yes, no matter which direction in space you move, you're moving "towards" the singularity. But a better way to look at it is that the singularity is a moment of time, not a place in space. You're moving "towards" the singularity in the same sense as you're moving "towards" next Tuesday. You can't stop moving towards next Tuesday by changing which direction in space you move. The same is true for the singularity once you're inside the hole's horizon.
Unless you went in butt-first, but the path of the photons would have changed and would now be going toward the black hole, and everything would look probably all smushed together.
You can avoid a coordinate, for example by choosing not to go there, or revisit another one repeatedly.
A black hole on the other hand doesn't have that: you cannot revisit old locations - attempting to do so moves you closer to the singularity.
The reason this phenomenon has a spooky-sounding name is that it also affects whether two objects can be causally connected. If you can only ever move closer to the center of the black hole, then there are (conceivably) other objects inside the event horizon that you can never have a causal relationship with.
But it doesn't mean that space and time literally switch places.
Also read Nick Gorkavyi: The Oscillating Universe: Einsteinian Cosmology of Black Holes and Gravitational Waves