The narrator wants to show that he is not insane, and offers a story as proof. In that story, the initial situation is the narrator's decision to kill the old man so that the man's eye will stop looking at the narrator.
The narrator goes to the old man's room every night for a week, ready to do the dirty deed. But, the sleeping man won't open his eye. Since the eye, not the man, is the problem, the narrator can't kill him if the offending eye isn't open.
This isn't much of a complication. The man has to wake up in order for the narrator to kill him. If the man still wouldn't wake up after months and months of the narrator trying to kill him, now that would be a conflict.
The narrator kills the old man with his own bed and then cuts up the body and hides it under the bedroom floor.
The narrator is pretty calm and collected when the police first show up. He gives them the guided tour of the house, and then invites them to hang out with him in the man's bedroom. But, the narrator starts to hear a terrible noise, which gets louder and louder, and…
Well, the noise gets even louder, and keeps on getting louder until the narrator can't take it anymore. Thinking it might make the noise stop, the narrator tells the cops to look under the floorboards.
Up to this moment, the narrator doesn't identify the sound. It's described first as "a ringing," and then as "a low, dull, quick sound – much such a sound as a watch makes when enveloped in cotton" (9). Only in the very last line does the narrator conclude that the sound was "the beating of [the man's] hideous heart!" (10)