You're only pushing to your repository, unless you have set up both your repo and digital ocean server to also auto-deploy on each commit, nothing will happen (thank god computers don't try to read our minds yet).
Basically, what you want is :
Whenever you commit, the repository should notify your digital ocean server that a commit was done. Your hook (on the DO server) will then check if the commit was on a certain branch (master ?) and if so: it will do a git pull etc. to update the server.
They will provide zero downtime deployment, run migrations for you, update dependencies, rollback in case of any problems, notify you via Slack, etc. etc.