To put an object to Storage, tap the icon with blue arrows in the bottom right corner of the screen, select to Storage (the icon with a green arrow and a box) and tap the object you want to put to Storage.
Tap the red cross in the bottom right corner of the screen to exit this mode.

Stored objects can be found in the very last tab of the Storage. You can place objects from Storage only on Big Land or Love Island. While on one of these Islands, go to last tab of the Storage and tap ‘Place' under an object you want to move out from Storage.