I've reloaded and retried a few times and checked the details of the situation. I have to kill a beggar in Ivarstead. He is standing by the destroyed house across the river. I go behind the inn, nobody can see me, my cursor says hidden. I get my bow and shoot, the beggar dies. I put away my bow, still hidden, my cursor eye symbol has not opened the whole time. Checking my menu reveals I have no bounty. With my bow sheathed, I walk out into view of the guards. They attack me and won't stop attacking me, even if I yield. The civilians are not attacking me, and will talk to me as per normal. Checking once again, still no bounty. Why are they attacking me?

Seems like you've got the case of the mystery witness. When you kill the Beggar, try leaving immediately and go far away, and not coming back in the area for a while, just think of it like if YOU were hanging around the scene of the crime, by logic you'd disappear as fast as you can. Also try and make sure you kill him without him being able to put up a fight. It usually prevents any possible alert.

Yeah, he was a one shot kill. Additionally, I was pretty far away and even came out into town on the opposite side of the inn to the side facing him, so as to remain inconspicuous, with my bow sheathed so as not to arouse suspicion. But still got attacked.