After installing this extension, look for the menu item "Compile on every file save" in the "Tools" menu. Clicking on it disables or enables this option (defaults to enabled). Your preferences is saved when exiting and starting Visual Studio again.
When enabled, the extension listens to your file save events. Every time you save any file when working on a Unity VS Solution now, this instantly starts compiling (building) the solution. This also means, that all your other files are saved as well, if they weren't.
When disabled in the menu, this extension does absolutely nothing in your project, so no worries here ;-)
The main idea where you would need this, is when developing for Unity. Usually, when saving your scripts, you switch back to Unity and have to wait until Unity finishes compiling until you can start working again. As this extension makes Visual Studio compile immediately, you save thousands of milliseconds in your workflow. If you are someone like me, who looks for a second at the code before switching back to Unity, it might even result in Unity already being ready for you when switching there.
(After I had a look at Rider for Unity, this was the main feature that I didn't have in Visual Studio. But not anymore.)
I am using this extension myself on daily basis. If you face any problem, don't be afraid to ask here.
Update v1.13: This script is now restricted to work with Unity VS Solutions only, as I had some trouble with it when editing a HoloLens (Windows Universal) solution. Feel free to comment, if you need this for more than Unity; I can insert another option for that as well.