Why does the narrator in The Tell-Tale Heart try to prove his sanity?

"The Tell-Tale Heart" is a short story by Edgar Allan Poe first published in 1843. It follows an unnamed narrator who insists on his sanity after murdering an old man with a "vulture eye". You can understand that the narrator was insane because he kept hearing the dead man's heart pounding.
He killed a man whom he "loved dearly"! He also "hears all things in the heaven and the hell". What sane person can do that? He is haunted by a "vulture eye" and so murders the old man in order to rid himself of the eye. Does that seem sane to you?