They may do it to escape a potential predator, as part of a courtship display, or to rid themselves of skin parasites. Nobody knows. It could even be a kind of play.
A smaller variety, called Mobula, jumps more frequently than any other kind of manta-ray. In the summer of 2005, Paul and Michael Albert documented these fantastic creatures at the sea of Cortez (Baja California), one of the largest concentrations of Manta Rays in the world. “The Flying Mobulas of the Sea of Cortez”, is the amazing article they wrote after their experience with hundreds of jumping rays.
“Whoosh! – wrote Paul - Without warning, a mobula emerges from below the surface, its long flat body glistening in the evening light and whip for a tail trailing behind. Flap, flap, flap, maybe a somersault or two, and then smack! It happened again and again. Single flips. Straight-up belly flops. Double flips. I see a single mobula leap a few times in succession; others leap only once and then disappear. I witness mobulas partially emerging from the water, one third of the wingtip still immersed, and rotate around that tip”.
Here you can see some of the few pictures and videos I could find. I hope you like them: