In some sense, nothing particuarly interesting happens at the event horizon of a black hole. For a sufficiently large black hole, the space at the event horizon is essentially flat [0]. The time and space swapping weirdness is entirely an artifact of the coordinate system being used.
If you think of spacetime as a 4 dimensional geometry, there is no particularly reason to pick out any specific shape of the "time" coordinate. For a given interval in space-time, there is notion of "proper time" that is invariant to coordinate system transformations, so a given observer does have an "ideal" coordinate to use for time. The "proper time" of a given observer could be the space axis of another coordinate system.
Assuming you have enough background in math, what I think is a reasonably understandable explanation is here:
[0] As with any gravitational object, there are tidal forces. However, due to the increasing radius of the event horizon, the strength of the tidal forces at the event horizon decrease as the mass of the blackhole increases.
If you think of spacetime as a 4 dimensional geometry, there is no particularly reason to pick out any specific shape of the "time" coordinate. For a given interval in space-time, there is notion of "proper time" that is invariant to coordinate system transformations, so a given observer does have an "ideal" coordinate to use for time. The "proper time" of a given observer could be the space axis of another coordinate system.
Assuming you have enough background in math, what I think is a reasonably understandable explanation is here:
https://www.youtube.com/watch?v=yv2-KPagCQs&feature=youtu.be...
Once he gets through explaining the hyperbolic coordinate sytstem, you can skip ahead to
https://www.youtube.com/watch?v=yv2-KPagCQs&feature=youtu.be...
Space is time line:
https://www.youtube.com/watch?v=yv2-KPagCQs&feature=youtu.be...
A somewhat clarifying picture of it taking "infinite" time to cross the event horizon, without anything interesting happening near the event horizon:
https://www.youtube.com/watch?v=yv2-KPagCQs&feature=youtu.be...
[0] As with any gravitational object, there are tidal forces. However, due to the increasing radius of the event horizon, the strength of the tidal forces at the event horizon decrease as the mass of the blackhole increases.