Releasing him, letting him sign as a minor-league free-agent with whomever he pleased, would also have accomplished this. Do the Dodgers still need cash?
Just judging by his usage/advancement, I'm guessing the Dodgers saw Bray as an org-depth fill in. Maybe they would have been fine to keep him in that role, or maybe they were considering releasing him in the near future anyway to open up a spot for someone else, and his name came up in conversation with the Twins and they let him go to pursue that opportunity, as a bit of reward for being a good organizational solider.
I don't know what that requires in return from the Twins -- a nominal amount of cash (like a 33rd round pick's signing bonus), or a little extra cash, or zero cash but unspecified "future considerations"? A question for Jack Goin, perhaps? But it seems to make the most sense, in this situation.