To respond to incoming SMS messages with Twilio and Java, developers need to configure a URL in their Twilio console that triggers HTTP requests when an incoming message is received. The response to these webhook requests should be written in TwiML, a XML dialect used by Twilio. This can be achieved using the Spring Boot framework, which provides a simple way to create an HTTP endpoint that handles incoming messages and returns responses in TwiML. By using the Twilio Java Helper Library, developers can easily generate TwiML responses without having to write the code from scratch. Once the application is running locally, it can be made public with tools like ngrok, which creates a public URL that forwards requests to the local server, allowing Twilio to receive incoming messages and respond accordingly.