As mentioned above, when scanned in-game, their RV appears and you can order themed furniture from it, but only 3 pieces at a time. Some amiibo figures have exclusive villagers that you can move in, like the Monster Hunter, Legend of Zelda, and Splatoon amiibo figures.
You can scan them once a day, every day, and they will give you a random item, sometimes one of their own, and 5 Meow coupons. Can do this with the amiibo cards, too. Cool, huh?
They're also compatible with Switch games like Zelda Breath of the Wild and Mario Kart.
People sometimes moan about amiibo but I do appreciate how Nintendo adds functionality to various games. Once you have one, it'll be useful in more than one game.
