The Americans are masters of mispronounciation. Many are unfamiliar with common words, and will pronounce them according to their own experience of how even more common words are spelled.
'Our' (belonging to us), and 'hour' (time) are pronounced in English with the 'ow' sound. So based on this experience, many Americans carry over this pronounciation to less familiar words like 'route' and 'tour'.
Working offshore, it took me some time to realise that workers 'on tower' were actually referring to their 'tour of duty'.
But note that this is a fairly recent phenomenon. When Nat King Cole sings 'Get your kicks on Route 66', he correctly pronounces it 'root'.
'Route', as in a road from A to B, is pronounced 'root'. So your wireless router, which is similarly establishing a roadway for signals from A to B, would be a 'rooter'.
(To add to the confusion, the name for a 'router', a power tool for shaping wood, is based on the verb 'to root', meaning to dig, to hollow, to scoop out. Yet these days this word is pronounced 'rowter'!). So it goes! ;-)