Why does the narrator decide to kill the old man?
The narrator is insane and believes that the old man has an evil force present within him, in his eye.
"for it was not the old man who vexed me, but his Evil Eye."
The Evil Eye is a belief that others had the ability to transmit curses with their eyes. Its purpose was to cast a spell of bad luck, disease or death upon its victim.
"It was open—wide, wide open—and I grew furious as I gazed upon it. I saw it with perfect distinctness a dull blue, with a hideous veil over it that chilled the very marrow in my bones; but I could see nothing else of the old man's face or person:"
The narrator may very well have been cursed by the old man's Evil Eye. After the murder, the narrator, who believes he has gotten away with the murder is haunted with a terrible sensation that he cannot ignore.
"My head ached, and I fancied a ringing in my ears: but still they sat and still chatted. The ringing became more distinct:—It continued and became more distinct:"
The torment that he faces grows and grows within him until he can no longer stand it.
"It grew louder—louder—louder! And still the men chatted pleasantly, and smiled. Was it possible they heard not? Almighty God!—no, no! They heard!—they suspected!—they knew!—they were making a mockery of my horror!"
The old man has beaten the narrator at his own game.
The narrator decides to kill the boarding house owner because of his strange, milky-colored eye (most like a cataract-covered eye). This eye becomes the obsession of the narrator and he decides that he must kill the boarding house owner. He boasts of his own cleverness and he stalks the owner. He watches him without moving an inch as he sleeps at night. The reader knows early on that the narrator is clearly mentally ill, obviously.
Once the narrator murders the owner, he dismembers his body and puts it underneath the floorboards. When the police come to question him about the murder, the narrator begins to hear what he perceives to be a heartbeat (probably the beating of his own heart). This causes him to become nervous and eventually paranoid. He finally cannot take it anymore and he screams that he killed the owner of the boarding house.
The narrator simply killed the owner for no other reason except the strange eye. The narrator even points out the the owner was a kind man who he got along with well.