Review your Purchase History here. Any amount you see showing in red or marked as Pending is a transaction that has not been paid yet. If you have any there, you can expect that to be processed when a valid payment method is added and you will not be able to use the App Store until that has been done.
View your purchase history for the App Store and other Apple media services - Apple Support
You do not need to pay anything to download free apps and you are not being charged to download a free app. When adding a credit card, you may be seeing a $1 hold to determine if the payment method is valid and it will automatically be removed before it is even posted to your account. If the $1 hold fails, then that card is determined to not be valid and you will need to use a different one.