Using Twilio Add-ons with Programmable Voice allows developers to combine pre-integrated third-party services into their applications, enhancing responses from Twilio by including additional data from partners. To use an add-on, developers must enable it in the Twilio Console, select recording methods, and set a callback URL. Once installed, Twilio includes add-on data in HTTP requests to the application, which can be accessed using various programming languages such as Python. The provided example uses Flask and the Twilio Python SDK to demonstrate how to access add-on data from an IBM Watson Speech to Text Add-on, which converts audio recordings into text using machine intelligence. By integrating add-ons with Programmable Voice, developers can build applications that provide more valuable insights and experiences for users.