To build a simple chat application with React and Nexmo's client-side JavaScript tooling, developers need to follow several steps. First, they must have a Vonage API account and create an application to generate conversation IDs. Then, they can add a new component to their React application at `client/src/Chatroom.js` and modify the existing `NexmoApp.js` file to use the new component. The chat room component will contain two states: choosing a chat room and the chat room itself, with functions for joining a conversation, sending messages, and rendering messages. Once the user is logged in and has joined a chat, they can send and receive messages using a textarea and button input field. The events raised by the message input are handled in `setInput` and `sendInput`, which store the inputted text in the component state and pass it to the Conversation using `sendText`. With this setup, developers have a basic chat application that can be extended with additional features such as error handling and performance optimization.