To demonstrate the setup of a Drone CI pipeline, a Cloudsmith account is first created and a new repository is generated. A `drone.yml` file is then configured with the necessary settings for authentication and package deployment to the private Cloudsmith repository. The pipeline is triggered by a push event to the repository, which triggers a build process that packages the application into a single executable file. The resulting package is then uploaded to the private Cloudsmith repository using the `drone` command-line tool, allowing for secure and automated management of the software package.