The Twilio SMS API empowers developers to programmatically send and receive SMS messages, track delivery, and manage message history. Auth0's Hooks feature allows developers to customize the behavior of their application with Node.js code executed at certain extension points, such as Post-User Registration, Post-Change Password, and Send Phone Message hooks. The post-user registration hook is used to notify users when they create an account by sending them a welcome SMS using the Twilio SMS API. To implement this, developers need to set up an Auth0 tenant, create a virtual phone number with Twilio, and then use the Auth0 Hooks feature to write custom code that sends an SMS to the user's phone number upon registration. The process involves importing the Twilio SDK, passing in credentials, and using the messages.create() API to send the SMS message. Once set up, developers can test their code using the Auth0 Runner tool and receive a response with a status of 200 and an SMS on their phone.