This is an idea introduced by psychoanalysts. I think it is a wise illustration of a common emotional and behavioural pathway that we all experience.
Here's an example:
Person 1 (in an irritable mood): "You're angry at me!"
Person 2 (in an neutral mood): "No, I'm not angry."
Person 1 (more angry): "Yes you are, I can hear it in your voice."
Person 2 (defensively): "I think I'm speaking the same as always."
Person 1 (more angry): "Now you're denying it!"
Person 2 (getting upset): "I'm not angry!"
Person 1: "Now you're shouting!"
etc. (the point here is that initially calm person 2 is becoming angry, due to projective identification originating from person 1)
In various emotional states, our emotions may strongly colour our perception of social exchanges. In "projective identification" our own emotion (most commonly anger) can cause us to perceive others as threatening. This perception may lead to an action. The action, especially if repeated, may cause the other person to actually become angry or irritated. From the initially angry person's point of view, the exchange appears to prove that his or her angry belief was correct all along.
This phenomenon causes the emotionally upset person to inaccurately attribute emotions to external events. Also it can lead to a vicious cycle in which stronger and stronger negative emotions are generated, "projected out" into the environment, then bolstered further by the consequences which follow. Also the upset person is bound to have more and more negative experiences, which further entrench the feeling of upset.
One of the tasks in a therapy environment is to gently consider the possibility of projective identification going on, and to prevent the vicious cycle from happening. This requires understanding, empathy, and a trusting therapy relationship. I think many such phenomena have to be pronounced, described, and discussed, in the here and now, in order for them to lose their power.