Another view on the past in “A Rose for Emily ” is that the shifts in time demonstrate one theme, which concerns the relationship between the new and the modern to the past and tradition. Through Emily and her house, the story shows the past as a repository of great human effort and integrity—after all, Emily is a “monument,” albeit “fallen,” and there is something quite wonderful about the town catering to her and calling her “Miss Emily.” However, the past is also a source of evil—Emily’s tyrannical father being one example, slavery another. The past has a romantic pull about it—we all indulge in nostalgia, but submitting fully to the romance of the past, as does Emily, denies the possibilities of the present and future, and in that way is a form of death, which we see in the horrifying discovery that she has been sleeping with the dead Homer for all these years to preserve the past. The narrator moving back and forth in time, therefore, signifies this push and pull relationship between past and present. The fact that the title of the story eliminates the “Miss suggests that the narrator frees Emily from the past by telling her story and explaining it in the present, yet he still preserves some of the beauty of the past by giving her a “rose,” a symbol of love, honor, and romance.
In addition to my colleague's answer I would add that the story jumps in chronology to make it seem more real. Faulkner seems to be using an extreme form of stream-of-consciousness writing wherein one recollection of the part of the narrator triggers the next memory. When a person tells a story, even if it doesn't span a fifty year period, he or she often jumps back and forth through time as one thought triggers the next. That's what the narrator does as he fits together the pieces of the puzzle about Emily. He jumps through time as he fits together each piece of the mystery.
I can almost picture the narrator(who appears to be an older man from Jefferson--old enough to recall the events of the past 50 years) saying between the text, "Hey, Joe, you remember that gawd-awful smell?--now it makes sense."
The unnamed narrator is a member of the town where Emily and Homer die. The reason Faulkner chooses to go back in time rather than from the time of the funeral is because the narrator, and by extension, the town, are figuring out what exactly has happened to bring about this tragedy.
The answers he finds are uncomfortable. Faulkner often uses the pronouns "we" and "our" to demonstrate the culpability of everyone who ignored and isolated Emily.
As the narrator gradually becomes aware of her mistreatment by neglect, so too do we as readers. It helps establish just how and why Emily went off the deep end. Still, it is quite a shock, not only to find Homer's corpse and discover Emily's death, but also to understand that she was so incredibly lonesome that she slept by his body all that time.
The force of the shock, I think, helps wake up the characters. For me, and a host of other critics and readers, the story would not be nearly as compelling if we did not understand the motivations of Emily and the neglect of the town before the discovery.