A trampoline is made up of a round piece of fabric (the trampoline 'bed') and springs that suspend the fabric from a rigid frame.When you give your natural jump in a ground just a little bit of a spring, defying gravity can came easy. But in trampoline, it provides you extra bounce in your bouncing. For a trampoline, it stores its potential energy in its spring (the energy stored in the spring is in the form of elastic energy)
In normal state, trampoline is in its equilibrium position and there will be no bouncing. In a standing state , your body (legs) exert a force on the trampoline that correlates to your weight. In that case also there will be no bouncing. But when you try to jump, the motion/acceleration caused by that movement of your legs is added to this(so the force is increased). And this force is exerted in to the trampoline. The trampoline stores that energy in its springs(as I told you earlier, it stores energy as elastic energy). Now Newton's 3rd law comes into action (it states that "for every action, there is an equal and opposite reaction). So the trampoline uses the stored energy in its springs to push back against you. After releasing the energy, the bed of trampoline comes to normal state. This is the overall physics of trampoline.