Docker 1.13 introduced enhancements to Docker Compose, simplifying deployment processes with new features in Compose V3, as explained by Alexei Ledenev during a Codefresh webinar. The session, available on demand, covers the ease of creating a multi-node Swarm cluster on a laptop without additional virtual machines. Docker's improvements, including Docker for Mac/Windows and the Docker Swarm mode from version 1.12, have made the deployment of applications on Swarm clusters more straightforward. Previously tedious steps involving certificates and console agents are now condensed into using simple commands like `docker swarm join`. The `docker stack deploy` command in Docker 1.13 allows easy deployment using a Compose file, which the Docker client can parse, unlike the python-based docker-compose CLI that overlooks the deploy section. The major enhancement in Compose V3 is the stack deployment feature for Swarm, demonstrated by Alexei, showcasing an application deployment on a single-machine Swarm cluster.