The first comment above hits the answer right on the head, but I'll give some more details.
The interaction between your foot and the soccer ball is inelastic - which shouldn't be surprising because all real-life interactions are essentially inelastic, but it's more elastic then a collision between say, a concrete block and your foot, because of the construction of the ball. The ball is essentially made to store elastic energy.
To study the energy of this system, let's call the work you can do on the ball (no matter if it's moving or not) $W$. So, if the ball is stationary, you give the ball energy $W$, some energy is stored in the ball elastically when it deforms, but when it returns to shape the energy is converted to kinetic. So the kinetic energy of the ball is $W$.
If the ball is moving towards you, with kinetic energy $K$, and then you kick it, what happens? Well first, you have to make sure you kick it hard enough for it to turn around, so roughly $K>W$ (you should actually do momentum conservation for this, but let's assume the collision is essentially elastic). The instant the ball is stopped, the elastic energy of the ball is $U=K+W$ - all of the energy has gone into deforming the ball. But when the ball returns to shape, all of the elastic energy goes into kinetic, and so the final kinetic energy of the ball is
that is, more then the energy you could give it when you kicked it from a stationary position.
Compare this to kicking a concrete block, which does not have the ability to store energy elastically. If you stop the block, all the energy $K+W$ goes into heating/destroying the block - or actually, destroying your foot!
So there are certainly some effects I am ignoring, like the fact that the interaction doesn't happen instantaneously and there are some sources of energy which I am not keeping track of, but in essence this is what happens.