How to enable saving on the cloud?
Under Settings>Apple ID, iCloud,iTunes & App Store (usually the top option) you need to have the iCloud Drive and Idle Animals enabled for your cloud save to be synced.
You need to make sure you have Google Play Games installed and updated before you install Idle Animals. If you still don't get the option to save to the cloud you can try rebooting your device and reinstalling Idle Animals, this will however mean you're likely to lose all previous Idle Animals saved data and /or purchases.
This is what will appear if Google Play Games needs updating.
Google Play games is installed and updated.
Idle Animals should appear in Google Play Games if installed correctly.