Video uses a completely different autofocus system (contrast detection using the main sensor, as opposed to the phase-detection autofocus points), so no, it's not to do with video.
39 AF points (or more in the higher-end professional-level bodies) means that there's a pretty darned good chance that something will be in focus. The question is what. When you let the camera decide, it might not always be what you were hoping for. Modern cameras are pretty good at keeping track of a moving subject they've already locked on to, even if it moves from AF point to AF point in the viewfinder. It's what they're going to lock on to in the first place that's the (potential) problem.
The 21-point autofocus uses only the very sensitive and accurate cross-type sensor group near the center of the viewfinder (the ones outlined with the larger, squarer markings in the viewfinder), and leaves the linear points outside of the center area out of the equation. When you let the camera decide where to focus, that gives you a greater degree of control—it allows you to ignore objects that are outside of the center area. When you use all of the AF points, the camera may decide that the foreground twig sticking into the frame over on the left is far more interesting and evocative than the subject you were hoping to capture—the camera is very much biased towards closest-subject detection. With just the center group involved, you can still let the camera do its best to find a focus point while still having a good degree of control.
The 9-point configuration gives you 3 points in the center group (top middle and bottom of the center row of the group) plus three points on either side, arranged in a diamond-shaped pattern. It becomes a lot easier to manage making your desired subject falling exactly on one of the sensors to attain focus or to manually select one of the focus points to use, but it usually means you need to slightly recompose the shot after focusing. That means it will be far less useful for moving subjects. (But not completely useless. 9 AF points is more than anyone had to work with only a few years ago. The D70, which occupied approximately the same market niche as the D7000 three camera generations ago, had an amazing 5 to work with, and I still remember when having 3 in a top-of-the-line professional film SLR body was a huge deal.)