The term "extinction meter" is great; the meter, not so much.
Basically, it works like this: on the camera above, the meter has numerals 1 through 7 in a linear scale of decreasing opacity. You point it at the subject and look through the window - the lowest easily visible numeral is then indexed against one axis of a provided chart. The other axis shows the available apertures, and the resultant point on the chart gives the appropriate shutter speed.
Sounds sort of simple, but it's complicated by several things. The reading on the meter is fairly imprecise, especially considering differences in people's individual vision. According to the manual, there are different "corrections" that have to be applied in different conditions (indoors, outdoors, etc.). If that weren't enough, the chart is indexed for ASA 32 film.
It's an interesting curiosity, but I don't really envision using it. Purely as a mental exercise, I might make up an adjusted chart for a modern film speed, but even so it's just not reliable enough to be useful.