You need to call the saveAsync method of the Office.Settings interface to persist the in-memory copy of the settings property bag in the document.
Any settings previously saved by an add-in are loaded when it is initialized, so during the lifetime of the session you can just use the set and get methods to work with the in-memory copy of the settings property bag. When you want to persist the settings so that they are available the next time the add-in is used, use the saveAsync method.