The Booster section of the Store and Inventory contains boosters: items that speed up game progress by increasing the amount of experience and coins you get, by increasing your chances of finding an item, etc.
Boosters that have a limited duration are usually activated from the Inventory.
Boosters that are used once per location are activated in a location or mini-game entrance window.
You can get them as a reward for completing investigations and Event quests or for assembling collections. You can also buy them in the Curiosity Shop.