How does Docker integrate with Kubernetes and what are the benefits of using both

How does Docker integrate with Kubernetes, and what are the benefits of using both?

Docker is very often used as a backend to Kubernetes, where Docker does the actual containerization, and Kubernetes handles the orchestration. How would Docker integrate inside a Kubernetes cluster, and what advantages does deploying with both Docker and Kubernetes, containerized applications, bear? Specifically, what features does Kubernetes offer in terms of running Docker containers at scale, including things like automated scaling, load balancing, and self-healing?