Bombes found the keys
Small correction - the Bombes didn't decrypt the messages themselves, but rather found the Enigma settings used to encode just one sample message on a particular network, a process that might take an hour or more. Once the settings were known much simpler machines could be used to decode the messages themselves very quickly (different keys were used on different networks and the settings changed each day so a number of bombe runs would be needed).
There is a good description of the bombe operation on http://www.ellsbury.com/enigmabombe.htm