The formation of an event horizon requires a certain level of density. This is when the black hole is born.

Inside the event horizon (it is postulated) matter gets compressed. When there are no forces to resist the gravimetric compression the center of the black hole gets smaller and smaller. (The center, not the whole region inside the event horizon). It becomes a point and forms a singularity. As it is a point it has no volume, yet it has some finite large mass, therefore its density is infinite.