NtwaiDmela: Ultima 7, DeusEX GOY Edition and its sequel
I think I know what the issue is. You're used to steam and Desura and other websites where you download a client, the game is run off that client, and there's little more than just downloading.
GOG.com does not use a client, and if you've downloaded anything off the site, it's the GOG-Downloader. The purpose of this is to keep downloads from being interrupted in the event of a loss of internet connection, reduces the likelihood of lost data, and supposedly results in a faster overall download.
However, what you download from the website is an instillation package. The entire game is on it, but it has to be unzipped, sorted and copied into your computer. If you're using the installer, then when the bar gets to 100% you click on the magnifying glass beside the game and that will activate the installer. If you don't use the downloader, you have to select to run or to save the install package, which if you simply run, it can fail to work.
After that, simply agree to the Terms and Services notice, click next, choose the folder to download to (Or ignore), click next, the installer will run, and a shortcut will be assigned to the desktop of your computer, along with a folder that will have access to all downloaded titles.