Poe could have told this story from a few different perspectives. The events of the story would have passed just about the same had Poe chosen to use a third person omniscient narrator. The difference would have been in how the reader judged the main character. That guy would have come across as a weirdo and nut case. He is a bit anyway, but Poe chose to use first person narration. By having the weird guy be the narrator, the reader immediately assumes the teller of the story is a good guy and reliable. It's not until the middle of the story that the reader starts to question the validity of the information being given by the narrator. But by that time the reader has already established a report with the narrator and is hesitant to change his or her initial judgment.
Another reason that I find the narrator convincing is that he has relatable personality quirks. Haven't you ever been bothered by some little odd character trait of someone that you know? In high school I was friends with a person, but I couldn't stand her laugh. It was like nails on a chalkboard. That's like the narrator and his master's eye. Of course I didn't kill my friend, so that makes me different.
He is also convincing because of how quickly guilt takes hold. Again, I can relate. Even if my action was an accident, if it hurt someone and I can't fix it, I feel SO guilty. And often I only feel better after confessing or apologizing. And that's what happens to the narrator. His guilt makes him hear the heart beat until he confesses. Poe effectively walks the narrator through thoughts and emotions that I can empathize with, so that makes the narrator convincing to me.