The new documentation site for a company has been built using Next.js and includes an embedded demo of their video chat feature, Daily. The demo allows users to quickly test out Daily calls and get a sense of what it would be like to embed the feature in their own app. To set up the demo, one needs a Daily account and can follow instructions on how to create a new repository locally and install dependencies. The demo uses Next API routes to create Daily rooms dynamically server-side, and includes a button that creates a room and starts a call when clicked. A callframe is created using a useEffect hook, which joins the room and listens for the 'left-meeting' event to reset app state when the local participant leaves the call. The demo also includes an input field that allows users to enter a room URL to join an existing room, and a "Join room" button that calls a function to set the room value in local state.