This frame-dragging is now being quantified. In a 2004 experiment which I can only describe as rad, glass spheres, perfectly round to within 40 atoms, were cooled to near absolute zero, launched into orbit, and spun up to create an exquisitely sensitive set of gyroscopes. The intention was to monitor the direction of spin of the gyros, looking for the tiny distortions caused by frame-dragging. However, a small manufacturing error introduced noise into the system, all but drowning out the telltale signature. Data analysis continues in an attempt to salvage the mission.
You wouldn't need super-sensitive gyroscopes to detect frame dragging near a rotating black hole. A spinning black hole is so massive and so dense that it drags spacetime around itself faster than the (local) speed of light.
(This next bit is written with a somewhat shaky understanding of relativity, corrections are welcome.)
As an observer directly approaches a rotating black hole, frame-dragging becomes more and more intense. The spinning current of spacetime threatens to sweep the observer into an orbit around the black hole, requiring more and more thrust to counter the effect as the observer approaches. At a certain point, all the thrust in the universe is sufficient only to keep the observer from being swept along, she is no longer able to fly against the spinning vortex of spacetime no matter how hard she tries. At this point the observer has entered the ergosphere.
At the outer edge of the ergosphere, spacetime rotates past at the speed of light. A photon precisely counter-orbiting the black hole at this distance would remain motionless. Further in, even light cannot fight the current. Past the boundary of the ergosphere, it is impossible to remain stationary relative to the black hole. Doing so would require the observer to move faster than the local speed of light.
The ergosphere is particularly cool because our observer can experience all this physics weirdness and later tell her friends about it. Parts of the ergosphere lie outside the black hole's event horizon, meaning that one can swoop in and out again without getting caught in any embarrassing and awkward singularities.