It slowly fills up while you are running (or matching tiles?). Once the bar is full, it's some kind of life saver: Whenever you are about to get defeated by a monster, the horn is triggered and all you captured monsters come to help you. They deal a lot of damage to all monsters on screen. It can't prevent you from failing to open up a chest, but when you run again afterwards, the bar is still completely filled. You can't trigger it manually.
One of the last shield upgrades has a passive ability to fill it up faster.
I actually never paid a lot of attention to it. You need to quickly clear your board to get more swords/staffs afterwards, or the next monster will probably still end your run. But it's really helpful if you are lucky enough to have a full bar when a hard boss appears.