Edit: have you tried simply clicking the icon again? Some people are having trouble getting GOG Galaxy to start on the first click. This may be because of a component that's crashing in the background.
If you're still having trouble, you can try a clean reinstall:
1. Uninstall GOG Galaxy from Control Panel or the Windows Settings app.
2. Open File Explorer. Copy and paste the following paths into the address bar and press Enter. Manually delete these folders, if they still exist:
C:\Program Files (x86)\GOG Galaxy
C:\ProgramData\GOG.com
%LocalAppData%\GOG.com
3. Install both
vc_redist.x86.exe and
vc_redist.x64.exe, then restart your system.
4. Install
setup_galaxy_2.0.80.33.exe which is currently the latest full installer.
5. If GOG Galaxy successfully starts up, click the cogwheel in the top left, choose Add Games & Friends, then Scan Folders for GOG Games. Point to the parent folder where you keep all your installed GOG games to add them again. You'll also need to reconfigure all of your customized GOG Galaxy settings.