My layman's understanding of a supernova explosion is that it's caused by a rebound when a massive star collapses. When the core collapses to a point where neutron degeneracy pressure dominates, the core collapse stops, and the matter falling in from the outer layers bounces back, causing a supernova (please correct me if I'm wrong). However, I've also read that the end result of the supernova can either be a neutron star or a black hole depending on the mass. When the star is massive enough to collapse into a black hole, what causes the supernova? What do the outer layers bounce off of? Shouldn't the whole thing just collapse into the black hole?
You are right. A direct collapse of a heavy star into a black hole can lead to a failed supernova without a detectable explosion.
However, a black hole can also form with an SN explosion. One possibility would be the initial formation of a neutron star and bounce off of the outer layers but subsequent fall back of enough material to collapse the neutron star to a BH.
In rapidly rotation stars the supernova can also be fueled by the energy released during the accretion of matter around the newly formed black hole. This would than be a Collapsar.
There is also the possibility for rapidly spinning stars that a magnetar forms during the supernova. Here, the explosion is not primarily driven by the bounce off and the neutrino wind but by the release of magnetic energy and a subsequent spin-down of the neutron star. As the rapid rotation can stabile a NS against its collapse, this spin-down can also lead to a delayed black hole formation. Here a paper about that.
So, SN explosions can become pretty complex and a black hole can form with and without a supernova explosion. All depending on mass, spin and metalicity of the progenitor star (& whether these is a binary companion).