A basketball bounces because gravity acts upon the ball creating motion or kinetic energy inside it. When the ball hits the ground, the kinetic energy is dispersed to the rubber surface of the ball, making it compress, and the pressurized air forces the ball back to its round shape.
The surface that the ball hits compresses as well. The deformed ball and the compressed surface act against each other. The surface pushes against the ball in order to get back its former shape, and the deformation in the ball pushes against the surface for the same reason. The energy of these two opposing forces create the bounce in the ball. Higher air pressure in the ball increases its efficiency and creates a higher bounce.