This is a question that is never fully answered by William Faulkner in his macabre Southern gothic short story, "A Rose for Emily." We can only assume that she killed Homer because he had decided to leave town and not marry Emily. Homer was her last, best chance at matrimony, and when she discovered that he had no intention of marrying her, she decided to keep him around anyway. Emily had a history of mental illness in her family, and she had already held on to one body in her house--her father's. She first refused to admit that her father was dead, refusing to give up his corpse after three days (in Southern heat), until a group of citizens persuaded her to give her father up, and "they buried her father quickly."
Homer received a similar fate, and it is possible that Miss Emily's mental condition would not allow her to recognize that he was dead. By killing Homer and secretly keeping his body in the upstairs bedroom, she could still enjoy his company--although in deathly silence--for the rest of her life. In her mind, Homer had never left her at all.
This is exactly what I was thinking. I am taking an online class, and sometimes I feel that my thoughts are way off base.