The power from your power supply normally powers both the fan spinning and the LED's.
However, spinning the fan is manually turning the motor (instead of letting the electric current doing it). Basically, energy is conserved. This means that the electric energy is converted into mechanical energy to make the fan spin. Vice Versa, mechanical energy can be converted to electric energy, such as through a generator. When you blow the compressed air over the fan blades to make it spin, this is actually turning the motor into a generator, which makes a small electric current usually refered to as Back EMF. That is why when you spin the fan, the LED's light up.
However, NEVER spin the fan blades on an LED fan. LED's are designed to have current pass through them in a certain direction. If you manually spin a fan, two bad things could happen. First, you could spin it so fast that too much current goes through and breaks the LED's. Second, and the more likely, the Back EMF generated could cause the current to go through the LED's backwards, which would make them pop. In fact, this Back EMF could actually go back to your motherboard and damage that as well.
This is why, with an LED fans, you always want to unplug it from your system and hold the blades when blowing compressed air on them so they don't spin. With non-LED fans, you should at least unplug those from your computer as well, because their Back EMF could damage your motherboard as well.