To make high-quality voice and video calls over the internet, businesses can use Twilio Flex, a cloud-based contact center platform that offers an intuitive and flexible solution for managing customer interactions across multiple channels. To enable VoIP calls on Twilio Flex, developers need to create a custom app with a soft phone embedded, a serverless app responsible for directing all outgoing VoIP calls, and a Programmable Contact Center that enables agents to receive VoIP calls. The process involves setting up a few accounts, downloading tools, and configuring the solution using a web-based demo frontend, Twilio Flex plugin, and TwiML app. Once deployed, users can make or receive VoIP calls in Flex, including inbound and outbound calls, with additional channels available for integration with other platforms.