I understand angular velocity increasing since the distance becomes shorter but why does the actual angular velocity increase? There is no force being applied perpendicular to the centripetal force so why does the velocity increase?
When the radius is decreased, there exists a component of velocity which is radially inward. The dot product of this component with the centripetal force is not zero, so work is done and speed increases.
Another way to think of it is that as an orbiting object falls deeper into a gravity well, it loses potential energy and gains kinetic energy, which means that speed increases.
Angular momentum is given by the cross-product of the momentum and the position vector. If one assumes these two vectors are orthogonal then the magnitude of the angular momentum is the product of the magnitudes of these two vectors. Hence if the angular momentum remains constant and the magnitude of one of these vectors decrease then the other one must increase. Since angular momentum in a closed system is conserved, a decrease in the distance would induce an increase in the momentum.