Why does time stop in black holes?
Time according to whom?
The fact is that, in special and general relativity, there is no universal time. Indeed, time is a coordinate in relativity so one must be careful to specify the coordinate system when asking questions like this.
Now, every entity also has an associated proper time which is not a coordinate which means that it is coordinate independent (invariant). Think of your proper time as the time according to your 'wrist watch'.
In the context of the static black hole (Schwarzschild black hole) solution, there is a coordinate system (Schwarzschild coordinates) that we can associate with the observer at infinity. That is to say, the coordinate time corresponds to the proper time of a hypothetical entity arbitrarily far from the black hole.
In this coordinate system, we can roughly say that the coordinate time 'stops' at the event horizon (in fact, there is no finite value of this coordinate time to assign to events on the horizon).
However, there are coordinate systems with finite coordinate time at the horizon, e.g., Kruskal-Szekeres coordinates.
Moreover, for any entity falling freely towards the horizon, the proper time does not 'stop'. Indeed, the entity simply continues through the horizon towards the 'center' of the black hole and then ceases to exist at the singularity.
We interpret the fact that the Schwarzschild coordinate time does not extend to the horizon as follows: no observer outside the horizon can see an entity reach (or fall through) the horizon in finite time. This is simply understood as the fact that light emitted from (or inside) the horizon cannot propagate to any event outside the horizon.
Why? Because the spacetime curvature at the horizon is so great that there is no light-like world line the extends beyond the horizon. Indeed, the horizon is light-like. A photon emitted 'outward' at the horizon simply remains on the horizon.
Within the horizon, the spacetime curvature is such that there are no world lines that do not terminate on the singularity - the curvature is so great within the horizon that the future is in the direction of the singularity.