Sticks, grass and leaves are used to expand your storage area. You can see how many you have by going to the storage tab in the bird inventory and tap the last storage slot which shows the storage expansion resources you have currently vs how many are needed to expand one slot. E.g. 13/1 means you have 13 of this item but only 1 is required to expand. You will need each storage resource to expand a slot and the cost will increase as you progress through the game. If you do not have enough expansion resources, you can expand using Gemberries. Sticks, grass and leaves are obtained around the forest as rewards for things like missions and achievements, in friend gifts, from the Bat etc.
Storage unlocks at a score of 300.