To handle incoming calls in CakePHP with Twilio, developers need to set up a new project, configure the database, create a model and entity, implement the controller logic, and configure the routes. The application will prompt the caller to speak their name and provide feedback, which is then processed and stored in the database. A dashboard template can be created to display all collected feedback, and the Twilio webhook configuration needs to be set up using ngrok. With this setup, the application can receive and respond to incoming calls, enhancing customer engagement and streamlining feedback collection processes.