First, you upgrade the source of FlaskBB using a simple
$ git pull
after that, the dependencies of FlaskBB should be updated (do not forget to switch into the virtualenv where you have installed FlaskBB into)
pip install --upgrade -r requirements.txt
If the database models changed after a release, you have to run the upgrade command:
$ flaskbb db upgrade
Don't forget to restart your application server when you are done upgrading!