Why is the Apple Store asking for payment for free apps?

You can download free apps, but you may need to have a valid payment method on file, even if the app you want to download is free. 

You can have a credit card / debit card on file and an Apple Account balance.

If you have set your Apple Account balance as your default payment method, your default payment method will be used first by Apple when you need to make a payment.

Free apps has also In-app pruchase.

See -> Buy additional app features with in-app purchases and subscriptions

Buy additional app features with in-app purchases and subscriptions

See also -> Apple Media Services Terms and Conditions

Legal - Apple Media Services - Apple

I think the app has the subscription method where in order to use the app you should get the subscription first. you can check it app in purchases below the app in App Store. Also, some users are kind enough to provide these kind of system in app reviews section be sure to check that.