In the very first paragraph of the story, the narrator himself answers this question. He states, "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." So, according to the narrator, (who isn't necessarily reliable, as parkerlee states) his hearing was the sharpest ("acute"), because of "the disease". So, we have to ask ourselves, "What sort of disease did this guy have?" According to eNotes, it was a "mental disorder" that caused him to have "delusions"; parkerlee also referred to this, mentioning that schizophrenia is a disease that often causes one to hear auditory hallucinations, and hence become delusional about reality around them. Poe never specifies exactly what disease it is, but we can guess and infer, based on the text. The narrator himself seems fixated on the idea that people think he is "mad" ("why will you say that I am mad?") and writes the entire tale to prove the is in fact not mad at all.
One can imagine him writing the tale while he is locked up in an insane asylum, and the writing is to prove or justify his actions. Too bad it doesn't do that; after reading it, we are even more convinced of the guy's unstable mental state.
The narrator's sense of hearing was not only very sharp; it was hallucinatory! Not only could he hear very faint sounds, but he also "heard" sounds which were not there at all. This is a clear sign of schizophrenia and/or hysteria, both severe mental disorders in which a person's perception of reality via the senses is distorted.
Even if you don't agree, this is not the only short story Poe wrote in which the tale is told by a madman or deranged criminal. (See also "The Black Cat" and "The Cask of Amontillado.") The beating of the "hideous heart" at the end of the story is only within the narrator's mind. This is what makes it so absolutely freaky.