The question is best answered in terms of elasticity.
A highly elastic collision will retain most energy (a good bounce), an inelastic collision dissipates (wastes) almost all energy.
Imagine a steel ball bearing dropped onto a thick steel plate. The ball bearing will bounce up and down for a long time - elastic collision.
Now imagine dropping a piece of plasticine.
One "goop" and it's all over - no bounce. Inelastic collision.
Putting more air into the basket ball makes it "tighter" and more elastic. The more elastic, the more energy is preserved, so the better the bounce.
For the same reason, pumping car tyres up to their safest pressure (and no more!!!) makes them more elastic, so they waste less energy when they deform while rotating, and you get (a bit) better gas mileage.
The more air pressure in the ball, the higher the ball will bounce. The less air pressure, the lower the ball will bounce. This is the same for just about any ball, too- not just basketballs.
When a basketball has more pressure it tends to bounce higher because of the consequent reduced surface area. The reason a basketball that has a lower pressure bounces not so high is because alot of the energy used to bounce the ball is lost because of friction and transfer to heat.
Surely the simple answer is that when filled to a high pressure, the ball is forced to retain a spherical shape (most volume for constant surface area). When such a ball hits a surface there is a distortion around the point of contact, and the higher the pressure the more rapidly the spherical shape is restored, which forces the ball to reverse its motion all the more rapidly. A deflated ball has no such pressure and won't bounce at all.
Also more probabilities are there for how air pressure effects on a basketball. The reasons may be as follows:
When a basketball is released, its own body goes down due to gravitational attraction. On the other hand, the air in the ball goes down after it as its weight is less than the ball. So the air will give an upward pressure which will effect the downward motion of the basketball. Again when the ball bounces up, a moment before that bounce, the air inside will start to come down quickly for repelling from the inside upward surface of the ball. Now when the ball goes up, the downward pressure of the air effects the ball.
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