If you look at the very opening lines, you can use inference, or educated guessing, to figure out why he is writing the story. He says,
"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."
In these opening lines, he seems pretty adamant in his insistence that he is not "mad," or insane. It almost feels like someone has accused him of being crazy, and he is defensive about it. He asks, "Why do you say I am crazy? I'm not crazy!! And, if you listen to my story, I'll prove it. I will calmly tell you what happened, and afterwards, you'll know that I am perfectly rational." He is defending himself against an accusation of insanity, basically. I can't blame the guy, although, his story doesn't really vindicate him; rather, it kinda proves that he is a bit looney. I hope that helped; good luck!