Under the Settings menu, go to iCloud, then select Documents and Data. Make sure it's turned to "ON". Under the Documents and Data menu, select on (or green) for your desired game and you should be good to go.

Backups tend to happen at save points, so make sure you play through at least one race in with an active connection to trigger the sync.  When you start the game on a new device with cloud syncing enabled, you should see a message box telling you of a save game conflict, and asking you to pick which game you want to keep.  Typically you should choose the game with more progress (on a new device, this should be the cloud game.)

You can store your game to the cloud and restore it after resetting or changing your device.