Why can’t I add apps on my new iPhone

Almost ALL free apps have In-App Purchases and for you download then, you MUST have a valid payment on your Apple Account. You never have to buy anything offered, but you do HAVE to have a payment method. So, if you want to download those apps, you'll need to follow the directions in the related article link at the top of this page, which brought you here to add a valid payment method.

You're not charged anything to have a payment method on your account. So no refund will be required. I suppose it's possible you could see a $1 charge to ensure the card is valid, but it really won't charge your card.

Will it immediately refund my money after I’m charged for the validation

You are not charged for validation. It will only take money if the app costs money to purchase it, or you make in-app purchases.