My impression is that only people in the bicycling social world believe that. It seems like a victim mentality that they reinforce by repeating it to each other. It's always possible I just haven't seen it, but localities around the country are building bicycling infrastructure, which doesn't correspond to hatred. Where do you see it?
I hardly ever hear someone expressing hatred of cyclists. People who ride obviously like it. The great majority don't care about it - it has little impact on their lives. In cities, on streets I see people honk at, yell at, and flip off cyclists just like they honk, flip off, and yell at other drivers. IME the cyclists generally 'drive' as well/poorly as the automobile drivers.
I do notice that people in spandex racing outfits on road bikes tend to behave with attitude problems toward everyone - pedestrians, non-racing cyclists, cars, etc. They are aggressive and fly by people, often with little margin, at dangerous speeds without warning. It's as if they think they own the road. I was just talking to a bike mechanic I know who brought it up. If people don't like them, it's obvious why.