@Astrochuck. For a long time, people said that winter “is cold and flu season” because scientists had observed that winter is the time when certain viruses predominate and spread. I will focus my answer on influenza, since it is an easier to study virus than “the cold virus”, which is actually hundreds of different viruses.
Here is a graph of the seasonal incidence of influenza A and B in the United States taken from the CDC website. On the bottom are the weeks of the year, and you can see that influenza predominates in December through March. In the Southern Hemisphere, influenza predominates from May until September (wintertime in the Southern Hemipsphere).
For a long time, people had speculated that the cold weather somehow allowed better survival of the virus, or perhaps better transmission, but direct data was lacking. Recently, a group at NYU showed (note: this technical article is freely available online) for the first time that cold temperatures and low humidity (i.e. dry air) enhance the transmission of influenza (the flu) in a guinea pig model of infection.
This article touched on three potential mechanisms to explain the role of low humidity in enhanced transmission. One, that the mucosa becomes dry (as has been mentioned above). However, owing to the short time frame of their experiments, they did not believe that the mucosa of the animals dried so quickly. Two, that low humidity improves the stability of the virus itself. And lastly, that low humidity favors the formation of smaller droplets of fluid, that can survive longer in the air (compared to wetter, larger droplets) and thus lead to better transmission.
As for the cold air, the authors speculate that “cooling of the nasal mucosa is thought to increase the viscosity of the mucous layer and reduce the frequency of cilia beats [8]. In this way, breathing cold air would slow mucociliary clearance and thereby encourage viral spread within the respiratory tract.”
In summary, influenza has become a seasonal infection in part owing to its relative success and transmissability during the winter months. Now that it has established that seasonal variation, it is here to stay (until global warming kicks in, then all bets are off).