I have plenty of coins and stars but unable to purchase buildings. Why can't I?

Once you have the required stars for a building, buy it with the necessary coins. When placing it on the island, you need an open 3x3 space, free of decoration like grass tiles. Occupied space will appear red, while empty spaces will be green, under the building you are trying to place.

If you are lacking space, place some low-flower items in storage. You'll lose some flower value, but you'll be replacing it will higher flower value, which will help you progress in the game.