I think Miss Emily killed Homer Barron because Homer had gossiped to the town that he and Emily had sex. It is said earlier in the story that after he began his construction work, Homer started getting to know everybody. It seems like he was a bright man with obvious charm, as it said he was usually in the middle of any laughing chatter or something to that effect, so it wouldn't be out of the realm of possibility that he told people around town.
Miss Emily would have been upset because she belonged to a family that was well respected, and Homer Barron is portrayed as a lower- to middle-class member of society, and a construction worker. But she would have been more upset that Homer didn't marry her. (Hint: it is said that Homer hung out at the club with younger men and didn't want to get married. He was a "player.")
Lastly, when Miss Emily goes to buy the arsenic to kill Homer, it says on the package, "For Rats." Homer Barron is a RAT, for letting the word out that he had sex with Miss Emily.
We have to do a whole thing on this for school, and that's my take on it. Opinions are welcome.