Adventurers can level up either by doing battle or through powering them up by using other adventurers as food. Powering up an Adventure gives that adventurer an emidiate boost in XP. The higher the level and the better the grade of an Adventurer the more XP that Adventurer gets.
When Adventurers are fed the same family of Adventurers as them they also obtain level ups in Abilities. (Example: When a Fire Goblin Spearman is fed a Water Goblin Spearman the original goblin will gain a random level up in one of its abilities.)
When using other Adventurers as food those adventurers get destroyed in the process. Be careful which adventurers you use for the powering up process.
Adventurers can be powered up through the tavern power up tab on the top of the screen.