Docker Compose simplifies building and running multi-container applications by using a single command to start a stack of containers defined in a declarative config file. It also improves the Docker container logs experience, allowing users to view unified log streams for all containers within a single terminal window. The docker compose logs command displays the log output from containers in Docker Compose stacks and offers several benefits such as easily correlating log output from multiple running containers, supporting compliance audits, and simplifying development experiences by live streaming logs straight to the terminal.