When you choose to download a '.license' file you just download it onto your computer, then store it somewhere safe for later on...
Your browser seems to think you want to open it and so it asks you for an app to open it - but don't ! just download the file...
You then open the licensing dialog from the extension's menu item in SketchUp and simply 'validate' the license there - choosing the saved license file as appropriate... This then links the extension to its license on that computer...