What is the science behind trampolines?

Trampolines use elastic collisions. The gravitational energy of the person bouncing is not sufficient to break the material, which rebounds to transfer energy back to the person.
It also uses the effect called resonance, whereby inputs of energy can be combined to form increasing amplitudes. By changing your body position (mostly by bending or extending your legs), you can increase the acceleration experienced before you hit, and increase your height before you start back down. Most of this is instinctive after a few minutes of bouncing.