There are several meanings to this phrase. The basic meaning is that instead of speaking out his feelings, one is acting them out. In its grossest form it means that a person is showing his emotions in the form of an overt action or behavior. One might slam the door, throw a book, crash a car, break a window, put a hole through a wall, spit on face, and what not to show his anger. Acting-out is an immature way of expressing one’s painful emotion. Children mostly act out. Adults are supposed to speak-out.
In its more subtle form, acting out means one is engaging in a purposeful, behavior, usually destructive to self or to others, as a way of dealing with his inner painful emotions and conflicts. A husband secretly angry at his wife might have an affair as a way of getting back at his wife. An angry boss might harass his underling by giving him a more difficult task. Often, this type of acting-out behavior has been labeled ‘passive-aggressive’ behavior, which means one is covertly aggressive toward another person instead of openly dealing with the underlying anger. When Princes Diana dated Al Fayed, the Egyptian, she was acting-out her anger toward her high brow in-laws.
Most acting-out people are not aware of the meaning of their behavior. When painful emotions are buried, they go out of awareness and they might manifest themselves in the form of a behavior. For example, if I am angry at a person and not know it, I might forget to keep an appointment with that person. Or, I might dump cigarette butts on his parking lot. Or, do some such stupid stuff as a way of expressing my secret dislike. Not until we become aware of our true emotions would we change our behavior.
Sometimes people act-out their feelings by being a champion of the downtrodden. Such constructive efforts in favor of worthy causes is called sublimation. Mothers grieving over the death of their children from drunk driving started the movement MADD. We can see around us numerous examples of people who have sublimated their inner emotions to achieve nobler goals.