Cole Sear: They see only what they want to see.
Cole Sear: I see dead people.
Cole Sear: Are you a good doctor?
Malcolm Crowe: Well... I used to be. I won an award once. From the Mayor. It had an expensive frame.
Vincent Gray: Do you know why you're afraid when you're alone? I do. I do.
Answer: Because as Cole tells us in the middle of the movie, "They only see what they want to see." The movie shows us Malcolm as he perceives himself. He can't actually put on an overcoat, or change clothes. He sees what he expects to see. He goes outside in the cold to visit Cole, so he sees himself wearing an overcoat. He doesn't know he is dead - he doesn't see his wound. To preserve this misdirection throughout the movie, we are shown Malcolm as he sees himself.
Actually, we do see Malcolm and Cole sees him. The entry wound in his abdomen is tiny, and the large, obvious exit wound is on Malcolm's back, which we never see.
C Classic