Connecting a GitHub repository to Cloudsmith allows users to automate the process of publishing and managing their software packages. Dan guides the user through integrating GitHub Actions, which enables automation of tasks such as testing, building, and deployment. By connecting GitHub Actions to Cloudsmith, developers can streamline their workflow, ensuring that their packages are up-to-date and easily accessible to others. This integration provides a seamless way for developers to publish and manage their software packages, making it easier for users to find and install the latest versions of their favorite applications.