how to download free app without payment method

You can have an account without a payment method if you have a NONE option in the payment methods. Otherwise, you need a valid payment method. You could buy and redeem an iTunes gift card as a payment method.

M1 Mac mini, macOS 12.0.1; Watch, series 5 GPS/cellular, watchOS 8.1; iPad Pro 2nd Gen, iPadOS 15.1; iPad Air 2, iPadOS 15.1; iPhone 12 Pro, iOS 15.1; TV 4th Gen, tvOS 15.1; HomePods, audioOS 15.1 and Airport Express w/ Airplay 2

You must confirm you have no app subscriptions including iCloud storage - if you do you must cancel or downgrade your storage to the one Apple provides at no charge.

Then assure your Apple ID is not associated with family sharing, which as an organizer requires you to have a payment on file that clears (yes, even for free apps or updating existing apps) - if you’re sharing purchases I’d make sure that is turned off as well.

If you’re still unable to remove all payments then most likely you have payments or purchases using your Apple ID that need to be satisfied prior to removing all payments on file.

Hope this helps. Most of these things can be accessed by opening the settings app and clicking on your iTunes account (located at the top: normally including your name & either an avatar/ monogram / profile picture)

Nothing that you posted here addressed what he said his issue was!