How is Emily a mystery?
Emily is a mystery because she lives in such a bizarre way and stays so aloof from everyone else in the town. Because of this, no one has any idea as to what is going on in her house, let alone in her head.
As far as we can tell, the only person Emily ever speaks too willingly is her servant, Tobe. When the town fathers come to talk to her about her taxes, she makes it clear she has no desire to really talk to them and they are soon sent on their way.
In addition to the fact that she doesn't talk to anyone, there is the fact of the bad smell that came from her house. No one has any idea as to why that happened or what was causing it.
So when you add weird things like the smell to the fact that she won't talk to anyone, Miss Emily becomes quite a mystery.
Specifically, Faulkner keeps Emily a mystery in "A Rose for Emily" by using the narrative point of view of the townspeople. The narrator reveals only what the townspeople know. Emily is secretive, so the townspeople know little about her. Only those events that are in some way public are, of course, known by the public.
The reader knows about Emily trying to keep her father's body after he dies, because the townspeople knew the father, knew that he died, and had to go into her house to retrieve the body. The same is true for everything else the reader learns about Emily.
Emily's a mystery to the town, so she's a mystery to the reader.
Faulkner makes Emily something of a mystery by the ways in which he always keeps her at a distance. No personal acquaintances are mentioned as sources of information. When she is seen, she seems intransigent and resolute, and she seems successfully to ignore the importunities of the townsfolk. Later, when she is together with Homer Barron, she is also seen at a distance. When the men sprinkle lime around her property and in the basement, she views them at a distance from an upstairs window. The ending is a surprise, definitely, even though the clues for the surprise have been carefully planted by Faulkner throughout the story.