Adding screen sharing to a Twilio Video application involves several steps, including checking for support in different browsers, getting the user's screen, and publishing it to the room. The process starts by setting up a new repository with the necessary dependencies, including Node.js, Chrome extension, Ngrok, and Twilio credentials. The script then checks if the browser supports screen capture and uses the corresponding API to get the stream. If supported, it publishes the track to the local participant in the room. The application also includes buttons for sharing and unsharing screens, which are toggled based on the user's permission. Once set up, the application can be tested by setting up a video chat with a friend or yourself across two browsers, where one can share their screen with the other.