Article: Why Do Figure Skaters Spin Faster when They Pull Their Arms In?

In a spin,

a skater starts spinning in place on the ice with her arms extended. As she draws her arms inward, raising them above her head, she spins faster and faster without exerting any force. What causes her spin rate to increase if she is not adding energy from her muscles?

Conservation of angular momentum

means that the total angular momentum of a spinning object does not change unless a force acts on it. Decreasing the distance from the axis of rotation decreases the moment of inertia by the square of the change in distance, so cutting the distance in half decreases the moment of inertia by a factor of four. If the moment of inertia decreases but the angular momentum stays constant, the rate of rotation must increase. As the skater brings her arms in, she spins faster and faster, going from about 2 rotations per second to 10 or more rotations per second.

~ Quoted from Associated Content

Of course knowing the science behind it doesn't necessarily make you a better skater, but it gives you a better idea of why your coach tells you to do certain things!