The farther a bouncing ball falls, the higher it bounces. This is because a bouncing ball, like any other object, accelerates due to gravity while falling. A ball dropped from 4 feet above the ground hits the ground with greater force and bounces higher than the same ball dropped from 1 foot above the ground.
As a bouncing ball falls, its kinetic energy increases because its speed constantly increases. When it hits the ground, this kinetic energy is converted into elastic energy as the ball deforms and flattens slightly. This elastic energy is converted back into kinetic energy as the ball bounces back up. These conversions aren't completely efficient, so the ball can never bounce as high as the height it was dropped from.
If air resistance is negligible, as it is for most bouncing balls, gravity causes falling objects to constantly increase in speed, which also means a constantly increasing amount of kinetic energy. This means more energy available to bounce back up with.
When a person throws a bouncing ball at the ground, she gives it a large amount of initial kinetic energy. This is why throwing a bouncing ball at the ground produces a higher bounce than simply dropping a bouncing ball.