A developer builds a custom video chat app using React and Twilio Programmable Video. The app has two parts: a React.js frontend and a backend written in any language or platform. The frontend handles all the video chat logic using Twilio's JavaScript SDK, while the backend generates an Access Token that grants authorization to users of the app. The developer clones the provided React starter kit, installs dependencies, and edits the App component to control whether the user sees the lobby screen or the video room. They also add methods for joining and leaving the room, updating the identity state, binding event listeners, and rendering the Room and Participant components.