Based on the Jenkins official image, this CIServer container is intended for its usage inside a Kubernetes cluster. It is able to manage the host docker daemon using its own Docker client. The Docker client expects /var/run/docker.sock to be mounted as stated in https://jpetazzo.github.io/2015/09/03/do-not-use-docker-in-docker-for-ci/. It does NOT use Docker-in-Docker.
Additionally, it can manage the Kubernetes cluster where it is deployed on via the kubectl tool. This way the access to the Kubernetes API is streamlined by using the official CLI.
This container is intended to run inside a Kubernetes cluster, so the Kubectl CLI knows how to authenticate itself without further configuration, but the image can be extended adding authentication for external clusters. In the GitHub repository, deployment scripts for Kubernetes can be found.
If you want to run outside from a cluster, you need to expose port 8080 to access Jenkins console and mount the /var/run/docker.sock volume so Docker client can be used:
docker run -d -p 8080:8080 -v /var/run/docker.sock:/var/run/docker.sock jlarriba/kube-ciserver