Diode connection in reverse bias
It is the presence of holes and extra electrons in the semiconductor material that make it conductive. At the join between the layers in a PN junction the holes and electrons join together and cancel each other out creating that depletion layer which is an insulator.

When you attach a reverse voltage you force the holes and electrons to go towards each other which fills in more holes in the depletion layer making it wider and a better insulator.

Attaching a forward voltage pulls the electrons out of the holes in the depletion layer till it is no longer an insulator.

There are a good description here with some illustrations.