"Stops of stabilization" in this specific context refers to a standardized CIPA test which determines a shutter speed where the image remains acceptably sharp. They then calculate the number of stops to 1/focal-length, which is a rule of thumb for getting sharp images from the 1950s. So if a 200mm lens produced a sharp image at 1/10s in the CIPA test, then that would be 1/10 -> 1/20 -> 1/40 -> 1/80 -> 1/160 -> 1/200 about 4.3 "stops of stabilization".
The results from the CIPA test don't really hold up to the real world though once you move beyond ~4 stops.
Edit: yeah, a speeding bullet caught at 1/5000th: <https://flickr.com/photos/hoohaaphotos/5587502201/>
So with this level of stabilization, you'll take a picture of a running cheetah at 1/25 as if it were 1/2000 only as far as the stability of the camera is concerned. So if you're not tracking the cheetah you'll get a sharp background because the shaking of your hands has been nullified, but the cheetah is still moving within the frame and still blurry.