Why does the narrator of "The Tell-Tale Heart" by Edgar Allan Poe say, "But why do you say that I am a mad man?"
Edgar Allan Poe's short story "The Tell-Tale Heart" is told by a first-person narrator, and the line to which you refer is found in the first paragraph, though your quote, as written in your question, is not quite correct. The narrator talks directly to us, and he begins his story this way:
True!--nervous--very, very dreadfully nervous I had been and am; but why will you say that I am mad? The disease had sharpened my senses--not destroyed--not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad? Hearken! and observe how healthily--how calmly I can tell you the whole story.
This reference to madness comes as the answer to an unasked question, or perhaps the response to an unspoken remark. The narrator in some way anticipates that his listeners (readers) might possibly think he is mad (crazy); however, before we can say it, he tells us we we would be wrong to make that assumption. It is the first of at least four such statements, statements in which the speaker dismisses the idea that he could possibly be mad.
The next reference to his non-existent (according to him) madness happens several paragraphs into the narrative. The speaker says:
Now this is the point. You fancy me mad. Madmen know nothing. But you should have seen me.
The narrator is admitting the reality that his listeners (readers) probably do think he is mad, but he takes great pains to detail all the precise and well planned things he does to carry out his intention to rid himself of the man--and therefore the eye. Those details actually help confirm our suspicions about the speaker's mental state.
The next reference to madness is spoken by the narrator just before he recounts the actual murder of the man.
And have I not told you that what you mistake for madness is but over-acuteness of the sense?
Here he actually tells his readers that they probably do think he is mad, but he endeavors to convince us that what we think is madness (something decidedly negative) is actually a blessing called heightened sensitivity (something generally considered to be positive). We, of course, have been reading his horrific, mad story and are not convinced.
Finally, the murder has been committed and now the narrator is going to try to convince us that he is not mad by demonstrating his cunning craftiness as he ingeniously disposes of the body.
If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body.
Obviously anyone who murders a man, especially just because he is "creeped out" by the man's eye, has got our vote as a madman, and only a madman would take pride in figuring out how to flawlessly conceal the body.
The narrator asks an important question in the first line of this story: "but why will you say that I am mad?" Ironically, despite his attempts to persuade us he is not crazy, the narrator spends the rest of the story proving why, exactly, we will say that he is mad.