I understand that you're being prompted to verify your Apple ID payment information when attempting to download an app. I'd like to provide you with some guidance on this.
What happens exactly when you tap on "Continue"? Are you able to successfully verify your billing information on the next page?
Take a look at this article that may be relevant to your issue:
If your payment method is declined, or you have a billing issue with a previous purchase in the iTunes Store - Apple Support