I'd say cyclist hate comes from pretty much the same place as vegan hate. The mere existence of such a class of people does folks' heads in, because it's showing them up, which they refuse to consciously acknowledge, so they project their feelings of inferiority (I'm not even a vegetarian, FTR).
Throw in all sorts of authority figures who are only too happy to foster a sense of aggrieved entitlement amongst their followers, and you get some prize-winning arsehole behaviour.