First of all, if your game is saved to the cloud you should transfer everything over to the new device. On most games cloud saving is managed with iCloud. On Beach Buggy Racing 2, you need to secure your account in-game with an email address and password.
If you do not have access to your old account or want to restart your game on a fresh device:
Make sure you are logged into the App Store with the same account you used when you originally purchased the item. Then go to the Settings / Support screen and click "Restore Purchases". Please note that only managed items like the Coin Doubler or Go Premium can be restored. Consumables like coin or gem packs cannot be restored.
If there are purchases you can't restore, please send a copy of your receipt to support@vectorunit.com and we will try our best to help you recover your purchases.