I don't understand how from 3 independent values per pixel (RGB) they claim to derive 200+ independent values per pixel. Unless they are assuming a smooth "image" (all pixels the same RGB), perturbed only by the color card? Not exactly a camera then

They're not claiming to get that many values per pixel, they're getting that many values overall for the medium through which light passes between the card and the phone. The idea light comes from a source (e.g. sun), bounces off the various colors of the card and thus produces hundreds of different spectra, those all pass through a medium, and land on the phone camera. So you're getting one measurement consisting of hundreds of RGB values that each represent intensity of different spectra, and you combine it all together to get a single spectrogram.

Did you read the actual article. They go into the method they use at length.