The Twilio Voice SDK JavaScript Quickstart has been modified by adding a UI dialpad to the application, allowing users to interact with an Interactive Voice Response (IVR) and make calls using their cursor. The process involves modifying the existing HTML file to include the dialpad elements, updating the CSS file with additional styles for the hangup button, and adding JavaScript code to handle user interaction with the dialpad, including making outgoing calls and handling incoming calls. The updated application now includes a functional dialpad that allows users to make calls using their cursor.