There's nothing really complicated with this application. It simply shows you your dynamic IP address once you launch it. Obviously you can get that information from your Control Panel or on one of the numerous websites that show IP adresses.
However, Router IP Address can be useful if you have multiple computers connected and want to know what IP address works for all. You can also use it if you have a connection issue. Check your IP address in Router IP Address and compare it to the one in your connection settings.
Router IP Address might not be the most advanced application ever, but for those of you that hate having to look around for their IP address it'll make things a little simpler.