The narrator himself isn't entirely sure why he killed the old man. In fact, he begins by admitting that he isn't sure "how first the idea entered [his] brain." He goes on to concede that he loved the old man and that there wasn't any particular animosity between them. The narrator claims he never wanted the man's money, and then he seemingly tries to convince himself that the old man deserved death because of his eye:
I think it was his eye! yes, it was this! One of his eyes resembled that of a vulture—a pale blue eye, with a film over it.
The narrator seems to stumble on a justification, using exclamations to mark his emotional response to discovering a "valid" reason for his murderous acts.
Of course, the true reason that the narrator kills the old man is because he is mad. In the first sentence, he seems to be responding to some accusation that he is not sane:
TRUE!—nervous—very, very dreadfully nervous I had been and am; but why will you say that I am mad?
He then references a "disease" which has made his senses even more acute. His protests that he is not actually mad are full of indignation, yet any rational reader will realize that sane men do not commit murder over the appearance of an eye. In the end, the narrator also "hears" the heartbeat of the man whom he has killed, which is further indication of his disconnect with sanity. He is so convinced that the now-dismembered man's heart is still beating that he confesses to his crime.
While the narrator believes that he has killed the old man in order to forever rid himself of the man's "vulture" eye, his actions are actually a reflection of his madness.