I read that the analysis of the CMB anisotropies
observed by WMAP has led to a value of the amount of baryons in the universe.

Can someone explain the chain of reasoning that leads from the CMB anisotropy to a prediction of how much baryonic mass there is??

See Anisotropies in the CMB, a webpage collection & tutorial by Wayne Hu at the University of Chicago. Basically, the angular pattern on temperature difference we see on the sky in the CMB ("anisotropies", WMAP image here) is actually a wave interference pattern. Baryonic & non-baryonic matter interact differently with the photons, and so they will change the pattern in a definable way that can be seen in the first 3 peaks of the angular power spectrum of the CMB (image and explanation here).

The specific webpage in Wayne Hu's collection which answers your question is here: Dark Matter Density. But you probably want to also see Radiation Driving Force and the Summary page. You can find them in his framed collection by going to his anisotropies page and selecting "higher peaks". But you can learn a lot by going through the entire tutorial.

__________________
The point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it. -- Bertrand Russell