Sometimes when opossums are cornered, they will simply keel over into a dead faint. Many predators, such as dogs, don't quite to know what to do with a dead opossum, and will walk away. If the predator does leave, the opossum will eventually raise up its head, look around, and if danger has passed, scurry off to cover.
Research indicates that "playing possum" is not really an act. Instead, the possum suffers a kind of nervous collapse. Since opossums aren't well equipped for fighting, their "delicate nervous system" has adaptated by creating an "over ride" mechanism that pushes them into a kind of temporary catatonia which, ironically enough, can help keep the possum alive.
The reason "playing possum" works is that the attack response in many dogs and other predator is also genetic -- it has nothing to do with being "angry" or being "hungry". The reason a fox will kill as many chickens as it can is that the jerky and fluttering movement of the chicken triggers a "mass murder" response within the fox. If chickens simply "played dead," like a possum often will, a vixen would pluck off one hen and be on her way.
Ironically, another animal that will occassionaly go into catatonic shock when cornered is the red fox. A fox bolted into a net will sometimes slip into a kind of momentary catatonia as it's nervous system shuts down, the body stiffens, and a kind of nervous shock sets in. Again, this is a kind of defense mechanism. It must work well enough, for it is employed by two very different animals that are very successful. The red fox is the most widely distributed carnivore in the world, and the possum is one of the oldest animals on the planet.