"TRUE! nervous, very, very dreadfully nervous I had been and am; but why WILL you say that I am mad?"
By starting the story with the narrator asking the audience to judge his sanity, Poe is creating an atmosphere of mystery and suspense. Immediately the reader begins looking for signs of true insanity. The result is a short story that is more psychologically oriented than externally oriented. As the narrator continues to defend his sanity by showing the rational thoughts and actions he had in dealing with the gray eye, the reader is made more and more aware of the mental disease plaguing the narrator. While there is much external conflict, the focus remains on the mental state of the narrator, and to the driving force behind the action. Through this focus, Poe is able to create a story that is horrific both in the physical action and in the mental implications, appealing to multiple audiences.
An interesting parallel can be observed between this short story and Poe's "The Black Cat." In both the narrator goes to great pains to prove to the reader that he is not crazy but rather confirms by his manner and speech that he really is.
"The Cask of the Amontillado" is another confessional narrative, but unlike the first two stories, the narrator here had a definite motive (revenge) and is not necessarily insane.
By beginning this story with an exclamation, Poe already has his audience's attention. What' more, he asks the reader to determine his level of sanity with the rest of the introduction. By starting out this way, the reader is now intrigued, and wishes to move forward to answer the question posed.
This introduction works a lot like ledes in journalism -- they are meant to grab the reader's attention with a "hook" of some sort that the reader then has to know more about. Poe has accomplished this well by opening the story in the way that he does.
Poe's stories tend to make the reader think. He wants them to question things, draw their own conclusions, and most definitely think about the supernatural; what exists, what doesn't, etc. The first sentence of "The Tell-Tale Heart" does just that. The narrator starts with an exclamation, pulling the reader in, and then goes to try and prove that he is not insane, that he is still a good person, what he did was not wrong, and the like.