The reason behind the townsfolk's fascination with Miss Emily, as well as with her decaying home, comes from a combination of Southern, post-war culture and our natural curiosity towards people or things who go against the system.
The Southern, post-war culture either bound together or separated even further the citizens who had to endure the pain of witnessing the South trying to rebuild itself. That feeling of community is very common among Southerners, and one of the very primary reasons for their concern for Miss Emily.
Then, there's also Miss Emily: a woman who once came from true Southern, aristocratic stock and who is now a faded shadow of what she once was. This woman not only looks like a relic from the past, but she boldly refuses to catch up with the present. In addition to her fearless assertions Emily is, per se, eccentric; she keeps to herself, does not socialize, and is only seen but not heard. She is known to have refused to give up the body of her dead father for fear of being alone, she is then seen going around town with a "Yankee", much younger than her, of much lower social status (compared to what she once was expected to marry) and even presumed to be gay. The actions of Miss Emily go beyond unlikely for an older woman with a name and history attached to a small town.
Basically, these are the qualities that often make other people draw conclusions and make assumptions that later on become town legends. The enigma of her personality is a huge part of what drives the fascination of the people. Equally, the history attached to her family name, including her once impressive home, automatically places her in a sort of "local celebrity" status, regardless of whether she is out an about much, or not; to see the "ancient greats" take the place of the common man is also fascinating. Moreover, her odd choices and behavior, so likely to be chastised by those of her kind, are hard to believe. For all of these reasons, it is impossible for the town not to wonder, ponder, invent, and imagine a typical day in the life of someone as, literally, fascinating, as Emily Grierson.