The narrator knows he sounds crazy, so he continually tries to convince the reader that he isn’t.
The narrator of the story is a madman. Like many madmen, he knows, but tries to convince himself and others that he is not. The main thing that makes him seem crazy is his fascination with the old man’s eye.
I saw it with perfect distinctness—all a dull blue, with a hideous veil over it that chilled the very marrow in my bones; but I could see nothing else of the old man's face or person: for I had directed the ray as if by instinct, precisely upon the damned spot.
He knows the reader will be skeptical about his obsession with the eye and his insistence that it is evil. This is one of the reasons he keeps telling the reader that he is not insane, and describes in painstaking detail how clever he is. He wants the reader to admire and respect him, not dismiss him.
Of course, the narrator really is mad. Anyone who kills his roommate because he thinks he has an evil eye would have to be. His guilt takes on a new form of madness. He is done in by his own delusions when he confesses after hearing the sound of the old man’s heart beating loudly long after he is dead.