Building a custom Lua plugin for the Kong Gateway involves creating a simple plugin that adds functionality to API services by automatically appending a custom header to responses. Kong Gateway, which operates on OpenResty and extends the NGINX server, allows developers to create plugins that manage requests between clients and services. These plugins can perform a variety of tasks like authentication, logging, rate limiting, and more, offering enhanced API management capabilities. Developers can use the Kong Plugin Development Kit and tools like Pongo to set up their environment, write, and test plugins efficiently. The tutorial demonstrates how to set up a development environment, write a custom Lua plugin, test it, and use Kong's Admin API to manage services and routes. Additionally, the guide emphasizes the importance of testing and validating plugins using Pongo's testing framework and maintaining clean code with static analysis tools. This enables developers to extend Kong's functionalities while ensuring robust API management solutions.