Sign in with Apple is Apple's login service. People sign up for or sign in to an app or a website with the Apple Account they already have, and skip forms and new passwords. It is a form of single sign-on, in which one identity provider, here Apple, confirms who the user is for many apps.
In an iOS, macOS, tvOS or watchOS app, the developer enables the Sign in with Apple capability for the app's App ID and shows Apple's button from the AuthenticationServices framework. When the user taps it, the system checks the Apple Account on the device, the user confirms with Face ID, Touch ID or Optic ID, and the app receives a user identifier and, if it asks, a name and email address. The user can hide the real address; Apple then gives the app a random address that forwards mail to it. Two-factor authentication is required. Websites, also on non-Apple platforms, use Sign in with Apple JS and a REST API between the app's servers and Apple's servers.
It is a capability that comes with the paid annual Apple Developer Program membership.