App store

The store will require a payment method that can be correctly charged to access the store services including downloading free apps and updating apps for 2 reasons.

1. Its either a new Apple ID, and has not yet used the iTunes store,
2. The payment method on file is not verified.

If the payment method stopped being verified for any reason (usually because it was unable to be charged for something), then it will require re-verification and will pop up the screen until its verified. If you have a pending purchase, it will need to be completed as part of the verification process.

It may also ask for payment method verification if you are Part of family sharing. It should not charge the Family Organizer's payment method however if you have enough iTunes credit balance to cover the cost of the item plus any applicable taxes.

If your payment method is getting declined, you'll need to contact your bank or credit card issuer to find out why.

Once the payment method has been correctly verified it can be removed, and it should not ask for it again.

See here: If your payment method is declined in the App Store or iTunes Store - Apple Support

Note that for verification, Apple may place an authorization hold on it. This hold is reversed shortly after verification is completed.

See here: About payment card authorization holds from the App Store or iTunes Store - Apple Support

Why?

Apple needs to verify you.

You need a valid credit card with your name on it, issued by your bank.

Note: Most debit cards are not supported.

FYI, updates and free apps are still free with a valid and accepted credit card.