To build a video chat application using Twilio Video and React with functional components, you need to create an App component that renders a header, footer, and a VideoChat component. The VideoChat component uses React Hooks such as useState, useCallback, useEffect, and useRef to store data, memoize functions, clean up effects, and attach DOM elements. It also connects to the Twilio Video service using the token and room name, joins a room, and displays participants' video and audio streams. The Room component is used to connect to the Twilio Video service, join a room, and display participants' identities. The Participant component uses React Hooks to attach video and audio tracks to DOM elements and clean up effects when the component unmounts.