Speeding up

In order to speed up on a swing you pump your legs. This probably came natural to you as a child when you first got on a swing and had no one to push you but the physics behind it is very simple.
Basically, as you are swinging when you pump your legs, you are raising your legs above your body's center of mass and therefore rising the swing height. As you continue to pump your legs, the energy that you burn to do so will increase the swing's potential energy and as you keep going, all that extra height adds up and therefore give you a wider swing arc making you go higher.
What causes the lift? As your legs are pumping and your head is being thrown backwards, you are pulling your center of mass closer to the bar at the top and this makes you gain potential energy which turns into kinetic energy. As you swing back down, your kinetic energy is now higher at the KE max and this is what causes you to get higher and higher. 
Because you are not pushing against anything, and nothing is pushing against you, you have to "create" your own energy. You're not actually creating energy, just transferring it from your legs into the swing.