The in-app purchases in this game work through your main Apple account, just like most purchases on Apple devices.
The game will ask you to confirm before you make any purchases, and you'll need to be signed in to your account to be able to buy.
Your account will need to have a card selected so that you can make a purchase.
A purchase will usually come through immediately once you've confirmed to buy, but sometimes it may take a little longer to come through (especially if you're using a card for the first time.) If it doesn't unlock immediately, please give it a few minutes and then check back again by going back into the store screen again.