The narrator doesn't like the old man because he has a vultures eye, the eye angers him. The narrator is standing at the door, watching the old man while he sleeps, waiting for the right moment to attack.
Oh how that old man's eye angers me.
The man jumped up because of a noise he heard. As the man sat up, the light from the lantern shined on his vulture eye. When the light shined on the old man's eye, it angered the narrator, so he attacked. He pushed the old man onto the floor, then he suffocated him.
He dismembers the body, then cleans up all of the blood so there is no mess.
I am house sitting for the old man while he is away on a trip.
Okay then, nothing seems out of the ordinary here.
He rips the floorboards up to bury the old man under them. After he buries the old man, he puts the floorboard back into place.
Here he is, I killed him. I will never have to see his terrible vulture eye ever again.
The police arrived at his doorstep because they got a complaint about a loud noise. The narrator tells them it was him having a nightmare. He invites them in to look around.
He thinks it is the heartbeat of the old man, coming from under the floorboards. He begins to panic, thinking the police will hear it too. The police, still having a conversation, don't seem to notice the increasing thumping noise.
When he invited the police in, he sat them right where he buried the old man. They have a casual conversation at first, everything is going well. But then, the narrator starts to hear repetitive thumps.
The narrator admits to killing the old man because the guilt was getting to him. That thumping noise, that he thought was the old man's heart, was actually his.