Implementing Programmable Chat with Laravel PHP and Vue.js involves several key steps, including installing dependencies, setting up Twilio's chat service, creating a controller to manage token generation, and building a view component that interacts with the chat service. The application allows users to join channels, send messages, and switch between channels. The codebase includes various methods for handling user input, loading channel data, and updating the UI. To test the application, start the development server, fill in the form, and open two instances of the browser to interact with each other.