The Wishing Fountain is the main unlock hub in Tomodachi Life: Living the Dream. As you care for Miis and build island progress, you collect Warm Fuzzies and use them at the fountain to make wishes.
Wishes can unlock new facilities, items, creation options, Palette House categories, island-life features, or other progress rewards. The player version is simple: the more you actively help your island, the more chances you get to open new parts of the game.
Do not think of the fountain as a shop where every choice is equally urgent. Early wishes matter because they decide what kinds of activities and custom designs you can start using sooner.