If a figure skater is spinning with her arms outstretched why will she spin faster if she brings her arms in?

The answer is related to the conservation of angular momentum. A figure skater will maintain approximately the same angular momentum during their spin (minus a negligible amount due to the friction of their skates and wind resistance). When they move their arms in, they will reduce their rotational inertia by reducing the distance of the mass of her arms and hands from the axis of rotation. In order to maintain the same angular momentum, angular rotation is increased. See the link.

Its called the angular conservation of energy. No matter what the skater's position the skater produces a certain amount of energy per second. When his / her hands are extended the distance of the rotation is larger. When he pulls his hands in the weight is unchanged. TO keep the energy at the same amount the difference has to be made up by increasing the number of spins per time unit.