To send an SMS from a React application using Twilio, it is recommended to create a back-end server that implements the Twilio REST API, wraps up your credentials, and sends SMS messages on behalf of your application. This approach ensures that sensitive credentials are not exposed to the internet, reducing the risk of abuse or unauthorized access. The process involves setting up a Node.js server with Express, installing the Twilio Node.js module, configuring credentials in an environment file, and creating API endpoints for sending and receiving data. On the client-side, a React component is created to handle form submissions, state updates, and interactions with the back-end server using the Fetch API or similar methods. By following this approach, developers can build secure and reliable SMS messaging applications without compromising security.