Question 1:
Where does the energy of the jump come from?

Question 2:
How does this demonstration resemble hydrogen fusion?

Answer 1:
The energy of the jump comes from the elastic potential energy stored in the deformation of the popper. The gravitational potential energy merely serves as the activation energy required to unleash the stored energy. In fact, the height of the jump does not change with the height from which the popper was dropped.

Answer 2:
During hydrogen fusion, the rest-mass energy stored in the hydrogen nuclei is released (analogous to the elastic potential energy in the dropper popper). However, in order for two hydrogen nuclei (two protons basically) to fuse, they come first overcome the very strong electrostatic repulsion they are going exert on each other. This requires them to start their approach with very high kinetic energy. In the stars (and our Sun), this very high kinetic energy is provided by the gravitational pull of the stars themselves. As the hydrogen nuclei fall towards the centre of the star (analogous to the falling dropper popper), they gain the necessary KE to allow them to come close enough to other hydrogen nuclei for fusion to occur.