Twilio's Programmable Voice allows users to purchase local phone numbers in over 100 countries and forward incoming calls to international destinations. To set up this feature, users can configure a TwiML Bin to manage the call flow, including forwarding calls to specific numbers. However, users must be aware of potential issues such as "high-risk" numbers that require explicit permission to call, and SIP protocol limitations that may cause early media content to be sent before the call is fully answered. By understanding these nuances and adjusting settings accordingly, users can successfully use Twilio's Programmable Voice to solve international calling problems.