The narrator wanted to prove that he's not insane with a story as proof. The beginning of that story starts with his decision to kill the old man so that the man's eye can't look at him anymore.
The narrator goes to the old man's room every night for seven days to kill him, but the old man won't open his eye. Since the problem is the eye, not the man, the narrator did not kill him.
On the eighth night, narrator accidently makes a noise, the old man wakes up with his eyes open. The growing sound of heartbeat getting faster and louder, seized the narrator with anxiety that it might be heard by a neighbour.
The narrator kills the old man and dismembers the corpse and then hides it under the bedroom floor.
Then police came to his house by his neighbour's report of someone screaming, but the narrator calmly invites them into his house and sat closely to the corpse. But, he begins to hear a terrible sound that grew louder and louder.
He could not take the noise anymore, so he admits his murder to the cops and tells them to look under the planks.