To build a simple chat application using Meteor.js and Twilio's IP Messaging, developers need to first install Meteor.js and create a free Twilio account. They then start by creating a Meteor project called "simplechat" and run the app straight out of the box. Next, they separate client-side and server-side code into two directories, "client" and "server", and move relevant code from simplechat.js to these new files. The application uses basic routing with Iron.Router to load content on the homepage, and defines a template for the home page. Finally, it adds the Twilio IP Messaging library to enable chat functionality.