An extinction meter basically uses different density steps to determine the lowest level of light available. A chart is then consulted to determine exposure. They generally use letters or numbers in a progressively denser series of steps. They were a step up from the old actinic meters, using paper exposed to light.
2 issues related to these meters relate to the ability of the eye to adapt to low light. One is the initial reading has to be made with the eye adapted to lower light, and the other is the eye becomes adapted to the point of compensating for lower light.
I've never used one, so can't attest to how well they work, or how off they are when the eye is outside the "adapted range" of usability. Reading the old Kodak manual aroused my curiosity, and left me wondering.