I FIXED IT!!!! So here's one possible solution that seemed to work for me:
1. Open the Don't Starve Together install directory (I've got it under "Steam\steamapps\common\Don't Starve Together\")
2. Open folder \bin\ from there
3. Right click on dontstarve_steam.exe and choose "Properties"
4. Go to "Compatibility"-tab
5. Push the "Change high DPI settings"-button
6. Tap the checkbox for both "Use this setting to fix scaling problems" and "Override high DPI scaling behavior". I've got the following options chosen from the drop-down menu:
You may notice that the Windows 7 compatibility mode is on, but I checked and that doesn't matter for this specific bug.