To integrate SMS into a Wix site, developers can use Twilio Programmable Messaging and Velo IDE. With a free Wix account and a free Twilio trial, developers can easily add SMS functionality to their site. The process involves enabling Velo, installing the Twilio npm module, creating a backend code file with Twilio functionality, and adding an event listener to a button on the UI page. Once completed, the button can be clicked to send a message via SMS. This integration allows for real-time communication with customers and provides a flexible solution for rapid development, making it easier for developers to integrate third-party solutions into their sites.