Monster Taming is a skill Septem will discover during the early stages of the European Chapter of his adventure. This appears as an extra skill in battles for you to attempt to tame the enemy that you fight. NOT all enemies are tamable and these ones will show a red X on them when you try to tame them in battle.
Some monsters are allowed to be tamed, these ones are easier to catch when you lower their health (HP) down, the lower the better. Unfortunately, the game is so easy and your character(s) are so overpowered that getting a monster’s HP low is often hard to do because you regularly 1-hit them.
What can I do with captured monsters..? Well, you can use them in ‘Spirit Fusion’ to gain better Monster Cards that you can equip on your characters for bonus stats. I recommend you catch a few of the same cards so you can experiment with ‘Spirit Fusion’ to earn some nice cards for equipping on your team.
Also, during battle you can go into ‘ITEMS’ and choose the ‘LIVING’ tab to choose a captured monster and summon it into battle. It has a timer bar and once it hits max it will appear and attack the monsters in battle. The monster summoned will last until the end of battle then you will need to use another card in the next battle, and so forth, i’d suggest keeping them for Fusion and Bosses.