Swings work by converting potential energy into kinetic energy, then kinetic energy back into potential energy, over and over again. The kinetic energy is the fast part of swinging; it's the speed you have as you rush back and forth. Potential energy is the high part of swinging. The higher you go on the swing, the more potential energy you have. Even though you're not moving at the very top of each swing, you can think of your height as your potential for speed. A physicist would say that your kinetic energy--that is, your speed--is "stored" as potential energy at the top of each swing. More height is more speed, just waiting to happen.
You can increase the energy of your swing in two separate ways. If someone pushes you, that increases your kinetic energy. By pumping your legs, you can increase your potential energy. By raising your legs at the top of each swing, you can raise the overall center of mass of your body, effectively raising the height of your swing.
Pump after pump, the energy you burn in your legs increases your swing's potential energy. That extra height will add up, giving you a wilder swing ride.