Trophies are rare treasures that you can decorate your home area with. There are many different kinds of trophies that you can gather from different sources. On every expedition island, you'll get a trophy as a prize once you complete the main storyline quests. You can also buy some trophies with Gems and earn some of them by completing timed event islands.
Trophies are fancy memorials from your travels and adventures and they will look great on your farm as decorations. Some of them even gifts you Energy. Once you receive your Energy, that trophy will be required to be recharged to give Energy again. You can recharge a trophy with duplicate trophies.