When you verify payment for free apps the payment verification is just to help prove it's you as the account holder.
When you verify payment there will usually be a small amount taken from your debit or credit card or Paypal balance to verify your credit card, debit card or Paypal account.
However that small amount of money should be returned to you within a day or so.
When you download free apps you will not be charged any money even though your payment method is on file with the Google Play or other app websites.
It's just there if you want to download paid apps.
But you can still use free apps if you want too.