Building GitHub Copilot extensions using Layer involves creating a server that handles HTTP requests and subscribing it to GitHub events, which allows AI-driven integrations to deliver platform intelligence directly into developers' coding workflows. The process is likened to building a Discord bot, where a server can be written in any language and is not limited to specific tech stacks. The server processes incoming requests, allowing for customization such as user authentication or data retrieval from databases. After creating the server, a GitHub App must be registered to define where GitHub Copilot should send requests. The tutorial highlights using tools like Python, Ngrok, and FastAPI to set up and expose the server to the internet, and emphasizes the potential for Layer to manage various AI extensions across platforms, simplifying the integration process for developers.