What the hell? This has kept people busy not for centuries but for millennia. Let's start with Aristotle: he explained this prohibition as due to the fact(s) that fava beans resembled genitals (Ari, we know where your mind was at), or they resembled the gates of Hades, because the plant is harmful, because they resemble the nature of the universe, or because they were not oligarchic (they were used for drawing lots in some democracies).
Although Aristotle practically invented logic single-handedly, for which I belatedly thank him, sometimes (like all of us) he didn't practice what he preached/invented. I strongly suspect that these explanations may be mutually exclusive. But one has gained more credibility than the others. No, not the one about fava beans resembling the nature of the universe (I had never noticed the similarity). The one about their being harmful.
There is in fact a disease called favism, which is a deficiency of G6PD- a condition not identified until the last century. As the name suggests, people with this congenital disease find fava beans toxic. Further, one of the foci of the disease is in Southern Italy, Crotone in Calabria. Crotone is the area in Magna Grecia (the ancient Greek colonies in Southern Italy) where Pythagoras set up his school. I think this is the most reasonable sort of explanation, as I have often thought that religious prohibitions against pork, say, may have been due to the possibility of the dreadful trichinosis.
For more on this philosophical-gastronomical-medical issue, you can check out this article, with some yummy recipes (obviously written by a non-Pythagorean).
(in the picture, a detail of Raphael's School of Athens, showing Pythagoras writing non mangiare fave, while Parmenides smirks on the right)