On the surface, the obvious answer is that Miss Emily has killed Homer Barron. The clues throughout the story reveal that Miss Emily buys arsenic, Homer Barron disappears, and a horrible smell then eminates from Miss Emily's house. The less obvious answer might be that the town suspected this all along but did nothing about it.
In the final two paragraphs of the story, the narrator never sounds surprised that what lies behind the locked door is Homer Barron's dead body. Faulkner writes, "For a long while we just stood there, looking down at the profound and fleshless grin." There is no apparent surprise in this statement. It is merely a statement of fact. Then, in the final lines Faulkner writes, "Then we noticed that in the second pillow was the indentation of a head. One of us lifted something from it, and leaning forward, that faint and invisible dust dry and acrid in the nostrils, we saw a long strand of iron-gray hair." Again, there is no great feeling of surprise among those who discover Homer Barron's dead body or over the fact that Miss Emily was sleeping next to him.
Faulkner often sets characters in places where the townspeople enable the main charater(s) to behave in a certain way or get away with foul deeds. In Miss Emily's case, the events preceding the discovery of Homer's body allow the reader to see that the town's lack of suprise comes from the fact they suspected she killed Homer all along but chose not to acknowledge it.
In the early part of the story, the narrator spends a bit of time describing the occasion when Miss Emily's house was leaking a foul odor. No explanation is given for the odor at the time. The townspeople are unable to get into the house to investigate. They finally sneak into the yard and sprinkle lime around the house to quell the smell.
...four men crossed Miss Emily's lawn and slunk about the house like burglars, sniffing along the base of the brickwork and at the cellar openings while one of them performed a regular sowing motion with his hand out of a sack slung from his shoulder. They broke open the cellar door and sprinkled lime there, and in all the outbuildings.... After a week or two the smell went away.
At the end of the story, the reason for the smell is revealed with the discovery of Homer Barron's body. He had obviously died in the house and rotted there, and during that process, the smell of it had stunk up the town. This also explained the reason that Miss Emily had bought arsenic. See this exchange:
"I want some poison," she said.
"Yes, Miss Emily. What kind? For rats and such? I'd recom--"
"I want the best you have. I don't care what kind."
The druggist named several. "They'll kill anything up to an elephant. But what you want is--"
"Arsenic," Miss Emily said. "Is that a good one?"
The discovery of the body reveals that Emily must have used the poison on Homer.