Round and Round We Go

If no external torque is applied to a closed system then the angular momentum $\vec{L}$ of that system remains constant, which is the case in this demonstration, (ignoring any frictional forces in the shaft of the chair).

Now $\vec{L} = I\vec{\omega},$ where $I$ is the moment of inertia and $\vec{\omega}$ is the angular velocity. When the woman pulls her arms in she is effectively reducing her moment of inertia, so for $\vec{L}$ to remain constant her angular velocity must increase.