In answering this question you will want to refer to the way in which Poe creates personas and characters and manages to enter their minds, telling their stories using the first person perspective, and thus giving us as readers privileged access into the madness, insanity and phobias of his characters. Certainly examples of his work that you will want to look at that are perfect demonstrations of this are "The Tell-Tale Heart," "The Black Cat" and "The Raven," though these is by no means an exhaustive list. I will respond to your question by talking briefly about "The Tell-Tale Heart."
Clearly, from the very beginning of this excellent story, we are invited to analyse the character of the narrator, whom, from the very start, is shown to be a profoundly unreliable narrator. Note how the story begins and the kind of claims that the narrator makes:
True!--nervous--very, very dreadfully nervous I had been and am; but why will you say that I am mad? The disease had sharpened my senses--not destroyed--not dulled them. above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad? Hearken! and observe how healthily--how calmly I can tell you the whole story.
Clearly the narrator feels that he is able to make such grandiose claims at the beginning of the story, and yet by insisting on such claims we only doubt his account and his veracity yet further. The idea that he is able to hear "all things" in heaven and on earth and "many things in hell" immediately presents to us the idea that he is mad, and his obsession with the old man's eye and how he presents himself only confirms our suspicions. Through the presentation of unreliable narrators, Poe contributed greatly to the development of psychoanalysis in fiction.