Navigate with your CLI to the directory where you've installed the script and enter:
$ cd /srv/users/serverpilot/apps/name_of_your_app/public
$ git pull origin master
If you get a Git error that files will be overwritten, you can do the following:
$ cd /srv/users/serverpilot/apps/name_of_your_app/public
$ git fetch --all
$ git reset --hard origin/master
$ git pull origin master
This requires that the platform was installed with Git and will update all modified, deleted and added files.