After a brief hiatus from violence, The Hound wasted no time in grabbing an ax (the only weapon available to him), stalking five men, and slaughtering them with a few swings in the Season 6 episode "No One." So why did The Hound kill the Brotherhood Without Banners? They weren't innocent in the matter.
In the previous episode, The Brotherhood Without Banners found The Hound in the village he had joined, under the leadership of The Elder Brother, a monk who saved his life when Brienne of Tarth left him for dead on a hillside. The Elder Brother was also a recovering solider who had found religion and was building a sept, along with other peace-seeking people. It was sort of like a Violence Anonymous commune that was spiritual, but claimed no loyalty to The Old Gods or The New. There, The Elder Brother nursed The Hound back to health, and The Hound's strength came in handy with the whole living-off-the-land thing.
But when The Brotherhood Without Banners rode through the village in an effort to raid it, they recognized The Hound. The Elder Brother claimed that they didn't have any steel, food, or lumber to spare, but The Hound knew The Brotherhood would return to steal the few supplies they did have. He suggested to The Elder Brother that they fight, but his new leader was resolved not to return to a life of violence. Unfortunately, that choice led to his demise.
While The Hound was out late chopping firewood, five members of The Brotherhood Without Banners circled back and slaughtered every single person in the village, including The Elder Brother, whom The Hound found hanging from the skeleton of the sept.
He grabs his ax, goes back into the woods, and tracks The Brotherhood swiftly, murdering the ones he finds, and learning the whereabouts of the rest. But once The Hound finds them, he learns they've already been apprehended by other members of The Brotherhood and they are strung up to hang. After a bit of negotiating, The Hound learns that their own men want to execute them for the atrocities they've committed. He's eventually allowed to kill two of the three men himself, though he is forced to hang them rather than butcher them.
Ultimately, this leaves us with what may be an interesting alliance between Sandor Clegane and The Brotherhood Without Banners — those who are left, anyway. And at this point, it's anyone's guess what that could mean for the future of the show and The Hound's storyline.