To set up automated testing for a Strapi API using Jest and Supertest, developers need to install necessary packages, configure Jest settings in the `package.json` file, create a temporary SQLite database environment specifically designed for tests, and write test cases to verify the functionality of Strapi API endpoints. The process involves setting up a controllable instance of Strapi that operates in a virtual testing environment without affecting the main project, using libraries such as Supertest to make HTTP requests and test APIs, and employing Jest to simplify and enhance the testing process for JavaScript code. By following these steps, developers can ensure the reliability and robustness of their applications through automated testing, making it easier to identify and fix issues before they become production-ready.