Watchtower will detect if there are links between any of the running containers and ensures that things are stopped/started in a way that won't break any of the links. If an update is detected for one of the dependencies in a group of linked containers, watchtower will stop and start all of the containers in the correct order so that the application comes back up correctly.
For example, imagine you were running a mysql container and a wordpress container which had been linked to the mysql container. If watchtower were to detect that the mysql container required an update, it would first shut down the linked wordpress container followed by the mysql container. When restarting the containers it would handle mysql first and then wordpress to ensure that the link continued to work.