Downward Comparisons
Social comparison theory argues that people naturally make comparisons to other people. And these comparisons often make us feel worse about ourselves or better about ourselves. As we generally prefer to feel good, we are prone to making downward comparisons, or comparisons that enable us to look down on other people. Moreover, research based on this theory also supports the notion that people are more negative towards others when they have been insulted or belittled, and that this can make people feel better about themselves (it can help restore self-esteem). In one study, when people were told they were unattractive (using fake feedback), compared to being told they were attractive, they rated others not only as less attractive but also less intelligent and less kind. Put succinctly, being insulted made people more likely to demean others.
Classical Projection
Freud argued decades ago that people cope with negative views of themselves by perceiving other people as having particularly high levels of that same negative view. Basically, say you are feeling dishonest. You are then more likely to see other people as dishonest, and this makes you, in a sense, feel more honest yourself.
Research supports this idea. In one study, when people were told they were high in anger, they were more likely to perceive another person's behavior as exhibiting anger. And, in doing so, they had less angry thoughts themselves.
Ego Threat
Researchers have discovered that it is threatened self-esteem that drives a lot of aggression. In other words, it doesn't really matter if people feel good or bad about themselves in general. What matters is that people, in the moment, feel worse about themselves than usual.