A few customers have raised this recently, where they open a document in a document library, and when they click save, it wants to save locally, so they then have to upload the document back into the library manually. A bit of a pain then…
So to get round this, on the server, Desktop Experience feature should be enabled. This is a windows feature in server manager, and will require a reboot afterwards;
File > Options > Trust Center > Trust Center Settings > Protected View;
From here things should work nicely.