We all do it. Assume. Imagine others are thinking things. Being certain others are feeling things. But you remember what Walter Matthau said about assuming in "Bad News Bears?" Did I just date myself? I wouldn't go so far as his explanation for the problems of assumptions, but I would say that when we project onto others, we are causing an unfair dynamic between the projector and the object of that projection.
The idea of projection is commonly discussed in the realm of paranoid ideation. In this arena, projection is used unconsciously and in profound ways to relieve the paranoid person of his/her poisonous, frightening feelings and experiences of their inner world. Somehow it seems safer to believe it is coming from an external source, but if you've ever worked with or spoken with a paranoid person, nothing about their projective process renders a feeling of safety for the paranoid. What is does is dually make the paranoid person feel unsafe inside and outside the self.
But what I am addressing here are those times, for example when person A is afraid to speak with person B, BELIEVING that person B will react badly. Person A is certain of it and thus avoids Person B. What I refer to above as unfair is that Person A doesn't give themselves the space to speak and Person B isn't given a chance to be themselves - reply as they will, which just might not be anywhere in the vicinity of Person A's concern. Think of the discussions that aren't had, that are avoided and the resulting challenges that ensue in all kinds of relationships as a result.
When this occurs in the consultation room from the patient to the analyst, it becomes very useful fodder for interpretations of transference and can prompt evocation of recollections of historical experiences and information as to how those experiences have become both internalized and might explain the fabric of later relational dynamics. Unlike in life outside of a therapist's office, when this projective behavior happens in the analytic space, there is room to explore and understand and see how this behavior is operating elsewhere, realizing the dangers of generalizations...and finding alternative solutions for these situations.
These projections are feelings we have about ourselves - they can become so strong that we convince ourselves that others MUST feel the same way. When we give others a chance, we do often find that our own feelings are matched, but, in my experience, when given the opportunity, others can prove to disprove our beliefs/worries/concerns. It can be scary, for it is a risk. At the same time, by speaking and giving another person a chance to respond in their own way, relationships can evolve and strengthen, even if the object of the projection disappoints in his/her response. At least there is genuineness there because you've made space for it.