When you harass someone, you bother them to the point of exhaustion. It’s not that each attack is so severe, it’s the constant small attacks that wear someone out. If you harass your boss for a raise, you might end up getting fired instead!
Harass probably comes from the Old French word harer meaning "to set a dog on." Think of someone being hounded when you use this word. You can harass your parents with questions or requests, but in turn, they can also harass you to clean your room! While this is annoying, it's much worse if you are harassed in the sense of being verbally abused. The bully at school who harasses some poor kid will be subject to disciplinary action if a complaint is filed.