Every modern payment card follows the EMV standard. This allows any EMV compliant card to securely sign a transaction hash with a private key held in the secure card element and send the signature back to the reader device. The transaction is then broadcast to the Radix Public Network (or your bank if making a normal fiat payment). The Radix network then verifies the transaction and signature in the usual manner and if it validates, you've made your payment.
Video demonstration: https://youtu.be/pMw7hQ6x2f0