We are not told directly in William Faulkner's short story "A Rose for Emily" exactly what is Emily's motive for killing Homer Barron. However, we (as well as the narrator, in the form of the townsfolk) are given clues from her character and behavior that can certainly help us make a conclusion.
The first clue is that Emily is a repressed woman. We know this because she is a spinster who is unable accept the changes of time, because of her stubbornness in accepting that her family is no longer powerful, and her weakness in accepting change of any kind because she seems to be completely infused in the ways of the Old South. To this, we add that her father is her primary figure of authority, that she is co-dependent to the wishes of her father and that, when he dies, Emily refuses to give up his body until she is forced to.Right there we see a profile of weakness and co-dependency that surpasses any rational behavior.
Moreover, the townsfolk narrator tells of Homer's behavior as a prelude of what will occur:
When she had first begun to be seen with Homer Barron, we had said, “She will marry him.” Then we said, “She will persuade him yet,” because Homer himself had remarked—he liked men, and it was known that he drank with the younger men in the Elks’ Club—that he was not a marrying man. Later we said, “Poor Emily” behind the jalousies as they passed on Sunday afternoon in the glittering buggy, Miss Emily with her head high and Homer Barron with his hat cocked and a cigar in his teeth, reins and whip in a yellow glove.
For a woman of Emily's pedigree and particular upbringing, it is quite hard to imagine her dating Homer. Something must be wrong. Either she is in denial like she is when her father dies, or she accepts Homer's behavior as normal and simply happens to go about her business.
Emily, however, is not that type of person. The Old South hardly ever lets go of the memories, the glory, and the beauty of the better days. Neither does Emily; that is the primary message that Faulkner intends to bestow.
Hence, we can only conclude that Emily kills Homer when he is about to abandon her. This, in turn, triggers Emily's natural co-dependence and fear, into action. Unfortunately, this action is to smoothly and secretly give Homer arsenic, and then keep his body the same way that she does with her father dies.
This is not a new behavior for Emily. This is a repetition of a former act. To her, this is not unnatural, nor weird. It is us, the readers and witnesses of her actions who, ultimately, will bring the final judgement. However, none of us does. Why? Because Emily's death is awkward but also brings about the metaphor of the death of the last person standing in the Old order of things.