In many ways this is the core mystery of the series. Twin Peaks fans have spent two decades analyzing and theorizing about the significance of this phrase.

I think it has a lot to do with the underlying mythos of the series, which is itself a matter of some dispute. People generally agree that there are at least two spirit realms, the Black Lodge and the White Lodge, and that there are corresponding Lodge Spirits that do not have physical bodies and therefore are limited in the ways they can travel outside the lodges.

An ongoing theme of the show is the desire of the spirits to travel in and/or affect the real world, and the ways in which they manage to achieve this. Some take possession of weakened humans, while others seem able to exist in various mediums such as wood, electricity, and running water.

My impression of the owls is that they are a physical manifestation of the watching eyes of these spirits. Everyone in Twin Peaks is a puppet, controlled or manipulated on some level by the lodges, and the omnipresent owls are one symbol of many that represent the strings that hold them (others include industrial machinery, traffic lights, and even coffee).

In the show, owls are a sinister presence (the Log Lady wants to talk to Cooper where they can't see her, for example) but not treated with the same gravity as the "true" lodge spirits (such as the Giant & BOB). This leads me to believe that they are more the eyes and ears of the lodge spirits than any meaningful representation or manifestation thereof.

This idea seems reinforced by this rather horrific image of BOB with an owl masking his face.