The narrator is trying to tell us that he is not mad. He says that he's just nervous and diseased but he can tell us the story with a lot of detail. The narrator wants to kill the old man because of his "vulture" eye.
The narrator wants to prove his sanity to the readers. The narrator is kind to the old man for the whole week before he kills him. Every night at the dead hour(12) he would watch the old man sleep. The narrator opens the lantern to see if the old mans eye is open for 7 nights strait. Every morning when the old man woke up the narrator asked him how he slept.
On the 8th night the narrator is walking in to the old mans room and wakes the old man up.When the old man woke up him and the narrator had a stand off for one hour the narrator is standing in the room and the old man is sitting up in his bed. The narrator opens the lantern and sees that the old mans eye is open. The narrator hears a low,dull,quick sound and the narrator thinks that the sound is the old mans heart. The narrator hears the sound getting louder and louder. The narrator has a new anxiety, he thinks that the low, dull,quick sound will be heard.
On the eighth night , the narrator kills the old man by suffocating him with a mattress.
The narrator dismembers the old mans body in the bath tube. He hides the body parts under the floorboards. The police arrive because an neighbor heard a shriek from the old mans home. The police searched the entire house and then the narrator invited them to come and take a seat. When the narrator brought his chair he put his chair over the old mans body parts. The narrator starts to hear the old mans heart as the sound get louder the narrator starts flipping out and thinks the police are laughing at him.
The narrator confesses to the murder because he cant stand the sound of the old mans heart beat. The old mans heart beat that the narrator is hearing is actually his own guilt.