It's simply to use: When you're on a page you want to create a shortened link for, just click the button in your toolbar or set up a keyboard shortcut. (Actually, you can set up a ton of shortcuts if you want, available in the extension's options, but a simple "Copy url" shortcut is the most obviously useful—maybe Ctrl+Shift+C if you don't use that for the developer view default.) The shortened URL will be automatically copied to your clipboard, and you still get all the nice stats tracking built into the goo.gl web site. You can also generate a QR code with a click of the button and view your history (if you sign into your Google account). If your shortened URL is ultimately going into some sort of sharing service, you can also enable one-click sharing for various services.
In all, it's simple, very configurable, and quick.
goo.gl URL Shortener [Google Chrome Extensions]