The .qgs is the QGIS project file and the .qgs~ is a backup of the project file. It is an XML file and it contains all the project settings like the path to the layers, the layer names, styling and such. No data is stored in the project file. This is unlike your word processor that stores everything in one file.
What the QField Sync plugin does is save your .qgs project to the export directory and copy with it your data files. I have tested the plugin with a Shapefile and with data stored in a PostGIS database.
What happens to your layers depends on the Project configuration under the QFieldSync menu. This allows you to specify how each layer is treated when exporting the data. This is handy if you have a QGIS file with lots of layers and you only want to use a couple of those on your mobile phone. Use Remove to exclude the layer from your portable project. For Shapefiles, you can either select Copy, Remove or Offline Editing. For the PostGIS database, it's Offline editing, No action or Remove.
The Shapefile works fine, as in, the plugin copies the data to the export directory if Copy is selected. As a reminder, a shapefile is made up of several files having the same name but different extensions like shp, shx, dbf, prj... The path to the data source is set to relative in the qgs file, something like './your_data.shp'.
For the PostGIS data, if offline editing is selected, the plugins packs the data into a Spatialite file but I couldn't get that to work as QField couldn't find the data. If No action is selected, the data source points to the actual PostGIS database which didn't work in my case as it's locally hosted. So at this point, your best bet is to use Shapefiles. One important point is to make sure that the data files are in the same folder as the .qgs file in your mobile phone storage.