Homer Barron was the foreman of a construction company that had been hired to do work on the town sidewalks:
The town had just let the contracts for paving the sidewalks, ... The construction company came with riggers and mules and machinery, and a foreman named Homer Barron, a Yankee--a big, dark, ready man, with a big voice and eyes lighter than his face. The little boys would follow in groups to hear him cuss the riggers, and the riggers singing in time to the rise and fall of picks. Pretty soon he knew everybody in town. Whenever you heard a lot of laughing anywhere about the square, Homer Barron would be in the center of the group.
Homer was a Northerner, and so not fully welcomed by the townspeople, as they were still smarting from the defeat of the South in the Civil War. Miss Emily accepts him, though. We can assume from the passage above that the townspeople were well acquainted with him, and that would include Miss Emily. Faulkner does not include the scene in which they formerly meet, but just says that they began to be seen together:
Presently we began to see him and Miss Emily on Sunday afternoons driving in the yellow-wheeled buggy and the matched team of bays from the livery stable.
It becomes apparent that the two are involved in a relationship, and the townspeople begin to assume that they will marry. Then Homer just disappears one day.....