Twilio's TaskRouter is being used by ThinkVoice, a company that provides communication tools for businesses, to manage their call center data and provide reliable and scalable solutions to their customers. The tutorial by Brian Coyle explains how to build a Salesforce-powered call center using Twilio's TaskRouter and Open CTI. This solution aims to reduce dependence on third-party solutions and leave as much configuration in Salesforce as possible, resulting in a more cost-effective and error-prone management headache. The demo demonstrates the ability to handle multiple channels of communication, including phone calls, SMS, and other messaging platforms.