The narrator in "The Tell-Tale Heart" begins by announcing that he is not mad and arguing that nobody could possibly claim that he was. He justifies his argument by pointing to the fact that he can hear everything that is going on—not only in the real world in which he is living but also in heaven and hell. His senses, rather than being diminished by supposed madness or disease, have actually been sharpened. He notes that because he can hear everything that is happening in the earthly world, as well as in heaven and in hell, he cannot possibly be mad but rather is more perceptive than ever before.
The narrator of course is obviously mad from the beginning, as Poe's narration makes clear. However, he gives this evidence stating otherwise as a means of trying to convince the reader that what he is saying is true and that he is able to hear and perceive unexpected and surprising things, not because of any inherent madness but simply because his senses have been heightened as a result of his "disease." The narrator seems to recognize, however mad he may be, that he is likely to be disbelieved and dismissed. As such, he provides some reasoning as to why his acute perception of the world around him should be taken seriously.