The Payment Request API allows developers to access payment credentials stored in digital wallets like Apple Pay, Google Pay, and Chrome. Stripe provides a Payment Request Button Element that securely tokenizes these credentials for a convenient checkout experience on mobile devices. The tutorial demonstrates how to add support for the Payment Request API to a React site using Stripe and React. It covers setting up a new React project, installing dependencies, creating a Payment Request object, checking if a digital wallet is available, processing payment details from the customer's wallet, charging the payment details in a Netlify function, and automating fulfillment with webhook events. The tutorial provides a demo application and source code for reference.