The whole works is basically held together by a little spring clip that fits just under the lip of the dial portion of the grip. I used a small screwdriver to pry the clip out of its channel. In the photo below, you see the clip after removal.
With that done, the housing for the shifter (at left above) just comes off (assuming you've already removed the screw that tightens the shifter to the handlebar), exposing the minimal inner workings of the mechanism. There are basically three parts: a small spring, a bearing, and the rotating mechanism to which the cable is attached. The spring sits in a little hole, and the bearing on top of the spring. When the grip is turned, the bearing slides into one of three holes in the rotating mechanism, either tightening or loosening the shifter cable, and thus shifting the hub.
In the photo above, you can see the rotating mechanism at right with the red indicator arrow on it. Below, you can see the spring in its hole, with the bearing resting on top.
That's about it. I haven't put the thing back together yet, but I'm guessing that making sure the cable and bearing stay in place while the spring clip is reattached is going to require at least one more hand than I currently have. I'll have to look into getting another one.
There's a fuzzy diagram here of the whole works, and if you combine it with the photos above, you can get a pretty good idea of how it all works. Apparently, though, this was not a very successful design and seems to have only been on 1960s bikes.