Go tap on the animal habitat that the pet is in that you want to store.
Click on the habitat and you will see what looks like a cat crate.
Click on that and you will see a picture of each animal in the habitat.
Check off the one you wish to add to storage.
At bottom a green ADD TO STORAGE will show.
Click that and the pet will be put in your cellar.
Go to your cellar and take the pet out you want by dragging him/her on to the habitat.