How does a black hole spin? This seems like a pretty basic question, but I couldn't find a duplicate upon quick inspection. From what I understand about black holes they actually contain no matter. All of the mass that maintains the black hole (ie. keeps space warped) is found in the energy bound in the space itself. I may not be stating this quite right, but this is basically what Kip Thorne states in "The Science of Interstellar". Therefore, it seems that to maintain angular momentum during black hole formation that space itself must "spin". This would imply that space is actually "spinning" even in the vicinity of the Earth (to a much lesser degree of course). Is this true?

As an aside, I don't take everything that Kip says as gospel truth. He states several times that Io is the innermost moon of Saturn in Chapter 7.
What we call a black hole is a solution of Einstein's equations of gravity.

We know several exact solutions: Empty and flat space corresponds to special relativity. Other known solutions include that of a point mass, a charged point mass, and a solution, known as the Kerr solution in which spacetime around a point mass is being "dragged" 

This last solution can be interpreted as corresponding to a black hole that would form if a rotating mass were to collapse. As you note, a black hole contains no matter: it is a vacuum solution of Einstein's equations. However a black hole does have properties like mass and angular momentum.

The shape of space around the Earth can be approximated the Kerr solution of a spinning black hole, with a radius of about 1cm. We would therefore expect some dragging of spacetime around the Earth. In 2008 the Gravity B probe had measured the gravity around the Earth and confirmed that the Frame Dragging effect predicted was present and its results were consistent with  General relativity. Frame dragging around the Earth is fantastically small and the Gravity B probe was working right at its limit of sensitivity. It causes a gyroscope to deviate from its spin by 0.00001 degrees per year.