In order to upload log files to Warcraft Logs, you need to install a client application. You can choose which uploader you want to use below. The Electron uploader will be replacing the Adobe AIR one eventually, but for now you can install either one.
To use the Adobe AIR uploader, you need to download and install Adobe AIR from Adobe's Web Site. Make sure ad-blocking is turned off on Adobe's site, or you will not be able to see the download link for AIR.
Once you have Adobe AIR installed, you can download the client here.