Twilio is software that enables voice, video and messaging online and within apps. In a nutshell, it provides a communication bridge between phone carriers and the internet.
A good example of the use of Twilio is transport companies that use apps like Lyft. Users (drivers and passengers) can communicate with one another directly through their phones over the internet without revealing each other's numbers.
Zendesk, a popular helpdesk app, also uses Twilio to enable users to communicate without revealing their information.