The battles you see on the map (also known as "Training Battles" in game) appear after you complete a Quest on that location. You'll see a preview of the monster you'll encounter in the next battle at this location.
When you win a Training Battle, you receive Monster Souls, Magic Items and Stars.
Monster Souls: imperative for Leveling Up
Magic Items: essential for upgrading your artifacts
Stars: compulsory for opening Chests at the end of battles
Each Training Battle has a different difficulty level, indicated by the number of Stars below. 3-Star battles are the most difficult, but they offer more significant rewards and more Stars to open Chests.
Every time you win a Training Battle, some other battles on the map are refreshed. Don't see a battle you're looking for? Play a few more battles first - one might appear later!
💡Learn more about: battles with extra rewards and Can I only play the battles I see on the Map?