The credit card number that you gave them acts as a guarantee against your failing to show up for your meal. If you are a no show, you will be charged the set amount for each person on your reservation.
When you actually have your F. lunch or dinner you will choose an appetizer, entree, dessert and beverage from the regular menu (or partake of the buffet in the case of H&V). After the meal, you will be presented with a bill. You can choose to pay for it anyway you want: cash, credit card or dining plan credits. You will then be given the voucher which allows you entry via the F. package entrance.
The requirement for the credit card is standard Disney practice for all Disney dining shows.