Mandrakes have bifurcated roots, and folklore holds that when plucked, their screaming will make you made. (Again, John Donne's reference is famous.)
Folklore of course being one of the cornerstones of fantasy fiction. Said folklore is probably the reason why occult people are into mandrake roots, but none of that was explained in HP. The roots acted like folklore (which isn't how they are in reality) and they did no ritual.