When you're out in the wild, the "Train" option opens four little menus with 4 tiers of fight difficulty. Winning 5 battles in Easy unlocks Normal, winning 5 battles in Normal unlocks Hard, and winning 5 Hard battles unlocks the Hunt battle, which is a high-difficulty mob specific to each zone.
F'lhaminn and the other innkeepers give you rewards each time you beat one of these Hunt battles for the first time, and the Garlean machine she mentions in your screen is the Hunt battle in the Mor Dhona region. Come back to her after beating it.