One of the themes in "A Rose for Emily" is the conflict between the Old South (pre-Civil War to about 1875) and the New South (1875 onwards), with Miss Emily representing what's left of the Old South, and the town itself, which is the narrator and a character in the story, representing the Old South becoming the New South. The town as narrator/character is partly still in the Old South but is moving steadily to the New South.
One aspect of the Old South is a respect for, a reverence for, pre-Civil War southern society, which included a wealthy land-owning, slave-owning class. The Grierson family, of which Miss Emily appears to be the last surviving member, was part of the pre-Civil War aristocratic stratum of society in Jefferson. The town, even though it has begun to move away from the Old South, still holds onto a vestige of respect for the southern aristocratic Grierson family and therefore looks at Miss Emily as a sort of monument:
Alive, Miss Emily had been a tradition, a duty, a care; a sort of hereditary obligation upon the town. . . .
Later, in the "Homer Barron" episode, when Miss Emily appears to allow Homer Barron, a Yankee and day-laborer, to court her, the town is scandalized by her behavior because it believes she is associating with someone well below her class:
She carried her head high enough--even when we believed that she was fallen. It was as if she demanded more than ever the recognition of her dignity as the last Grierson. . . .
The town, still clinging tenuously to the Old South conception of an aristocracy, cannot believe Emily is abandoning her duty to act in accord with noblesse oblige, her obligation to behave as a southern aristocrat, and the appropriate behavior did not include an attachment to a Northerner and someone below her class. She is, to the town, fallen below her station in life.
In essence, then, Miss Emily is a monument because she is the last representative of the southern aristocracy in the town, and even though her economic circumstances are now grim, she is still accorded some respect simply for her family's history in Jefferson. She becomes a "fallen monument" by violating the town's social norms when she associates herself with someone not even remotely close to her class.