When a user needs to make an international call from their landline or local phone, they can create a custom international calling card with Twilio. This is achieved by setting up a U.S. number and forwarding it to an international destination using the Dial verb. The application requires several files, including handlers for incoming calls, PHP handlers for taking keypresses, and files for each of the phone menu options. These files work together to connect the call and direct the user to the correct number based on their selection. The system is pay-as-you-go, meaning the user only pays for the amount they use, eliminating the need for prepaid cards.