Free app asks for a payment to use it?
Yes. In-app purchases are secure just as they are on the actual App Store. Apple provides developers the APIs needed for in-app purchases. My understanding is that one can do purchases in apps through the "official" in-app purchase system.

You should be able to easily tell the difference between a normal in-app purchase, and a third-party one (which should, but may not be secure). For the official in app purchase, it should prompt you (system type prompt) with the amount of payment and option to cancel. If you choose to pay, it should prompt you for your Apple ID password. You should not have to enter credit card information, or other personal information (if you already have that info tied to your Apple ID account.