It may come as a surprise to learn that the membrane keyboard is one of the oldest designs of keyboard, going back to 1968:Datanetics elastic diaphragm array
However, the original design was far more elaborate than anything you'd see today. The following presentation was provided by its inventor, Meryl Miller:
This is a better photo of the layers:
So, why so complicated? I am not sure, but if you recall, it was commonplace in the 70s for keyboards to emit ASCII codes. This way you could theoretically mate any computer and any keyboard. Take the Apple I: it didn't come with a keyboard. Since you had to use an off-the-shelf keyboard (and it's been suggested by Mike Willegal on his website that Steve Jobs recommended that of Datanetics, using DC-50 switches), off-the-shelf keyboards needed to transmit keystrokes in a recognised format, and this format was ASCII codes.
From what I can tell, the Datanetics membrane system caused each keystroke to sequentially register two separate signals, using different membrane thicknesses to guarantee the sequence. This signal sequence somehow made it easier for the circuitry to determine which key was pressed, since I assume that in 1968 it wasn't considered feasible to place a microcontroller inside as became commonplace later in time.
This is really all I know. The patent doesn't make the operation patently obvious, as is so often the case, and I don't have any more specific details on the operation of this keyboard.
By 1976 (the date on another brochure), the design was greatly simplified: PCB with gold-plated contacts, spacer membrane, circuit membrane, protective cover. I assume that advances in technology made the impossible-to-understand original design redundant!
This design may have spawned both the modern membrane keyboard and
the switchplate system.