What happens to time at a black hole's edge, seen from outside?

It appears to stop. An object falling towards the event horizon is seen from a distance to slow, redden and dim, taking longer and longer to fall the last small distance, and it never quite arrives - the light it emits is stretched further and further until it fades out. From outside, nothing ever crosses.

From the falling object's own point of view nothing whatever happens at that moment. It passes through the horizon at speed, its clock running normally, with no local sign that a boundary has been crossed - the horizon is a location rather than a surface. Both descriptions are correct, and reconciling them is a large part of why black holes are interesting rather than just heavy.

That mismatch is also the setup for the hardest open problem in the subject. Hawking showed in 1974 that black holes should slowly radiate and eventually evaporate, which raises the question of what happens to the information about everything that fell in - quantum mechanics says it cannot be destroyed, and the radiation appears to carry none of it. Physicists have been arguing about the information paradox for fifty years, and the argument is one of the main reasons anyone believes gravity and quantum theory must eventually be reconciled.