This is a straightforward question which even the narrator cannot answer with clarity. He isn't sure how the idea of murder first came to him and admits that he "loved the old man." The old man had never committed any transgressions against the narrator, and he didn't want the man's money.
The narrator seems to stumble upon his reasoning for murder almost by accident:
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. Whenever it fell upon me, my blood ran cold; and so by degrees—very gradually—I made up my mind to take the life of the old man, and thus rid myself of the eye for ever.
With the narrator's inclusion of thinking it was the eye that drove him to murder, followed by the exclamation, it seems that he is trying to convince himself of his motives. Of course, people don't murder others because of an odd-looking eye, so all of this together can only mean one thing: the narrator is not mentally stable.
Therefore, the real reason for murder is that these are the actions of an insane man. Although he tries to convince his audience from the first lines that he is sane, there is no support for sanity in his actions. In fact, his subconscious eventually catches up with him in the end, and he admits to the murder because he is convinced that the heart of the man he has dismembered beats loudly underneath the floor where he is hidden.
The narrator's unstable mental abilities are the ultimate source of his desire to murder a man whom he claims he loved.