They figure if you are having the problem then there is a chance others might be having it as well. The log files tell them what the game is doing and not doing with the servers and might be able to provide additional insight into your problem. Also providing them with the log file allows them to get this information without you needing to know much about computers. This in turn helps them figure out what exactly is broken in the patch.
The two paths he listed above (one for Windows XP and the other for Vista) show where you can find the file named ImpulseReactor.log on your system. You can browse there via My Computer (or Computer in Vista) or probably by going to your My Documents folder then My Games, Gas Powered Games, and finally Demigod. You should see the indicated log file there.
Then just attach the file in an e-mail and send it off to support@stardock.com