As you have no doubt discovered the 25 inventory space you are initially given for your Hatchery is not much and you will want to upgrade it to so you can house more birds. Inventory upgrades are not expensive, you can purchase 1 spot for 25 Gems or 10 spots for 200 Gems. If you do not want to use Gems because you are saving them for something else you can always evolve your birds, and fuse other birds into your favorite birds in order to make space in your inventory.