Pretty simple actually. The trampoline mat or whatever it's called has a rack and pinion below it, the pinion turning a spring, the rack mounted vertically with the appropriate amount of padding necessary because you're jumping on a piece of metal.
When you jump on the trampoline, it pushes the rack
down turning the gear / spring assembly but bounces up in a reset motion keeping the energy imparted to the spring due to a one way ratchet.
That ratchet is adjusted to release after a set amount of energy is put into the system. So jumping say ten times, all that energy you put into it releases and shoots you into the air probably breaking your legs or killing you or both.
So a couple of bugs to work out, the whole death thing, but the concept is sound.
Actually, here's a simpler version: It's just an air bag underneath that fills a compressed air tank with each jump using a one way valve. At a set pressure the one way valve takes the last push of air from the jumper but then unlocks releasing all the stored air. This would be a cheaper way to make this that would be no the less fun / dangerous.