How does a remote server with docker images know when to pull a new image built with a CI tool?

We have been moving our project to automated deployments recently. For the past few days I have configured our docker containers to communicate with each other using docker-compose. Now we are moving everything to Gitlab CI, that builds the application image on a push.

However, Im stuck on a conceptual part of the flow. How does the remote Digital Ocean droplet that I have my application running on in coordination with the other docker containers know when to pull a new image built in CI?

Is there a script that needs to be written that checks the new image and pulls the container?

If I'm running the containers on the remote server with docker-compose, would the containers be able to be swapped out easily with the new container?

EDIT

In the gitlab docs the

deploy:
  stage: deploy
  script:
    - ./deploy.sh
  only:
    - master

What would deploy.sh look like?