This is a rather tentative answer since the evidence still looks fairly contradictory to me, but the BBC has an article that reiterates what DW says, with more detail:
Thanks to the laws of thermodynamics, cold air can carry less water vapour before it reaches the “dew point” and falls as rain. So while the weather outside may seem wetter, the air itself is drier as it loses the moisture. And a steady stream of research over the past few years has shown that these dry conditions seem to offer the perfect environment for the flu virus to flourish.
And with more details in the why part
Any time we splutter with a cold, we expel a mist of particles from our nose and mouths. In moist air, these particles may remain relatively large, and drop to the floor. But in dry air, they break up into smaller pieces – eventually becoming so small that they can stay aloft for hours or days. (It’s a bit like the mist you get when you turn a hose pipe to its finest spray.) The result is that in winter, you are breathing a cocktail of dead cells, mucus and viruses from anyone and everyone who has visited the room recently.
But insofar, it's still "just theory". The supporting experiments mentioned by the BBC:
Lab experiments, for instance, have looked at the way flu spreads among groups of guinea pigs. In moister air, the epidemic struggles to build momentum, whereas in drier conditions it spreads like wildfire.
This is practically an experimental contradiction of the epidemiological study from my question! And then come the qualifications:
There are some exceptions to the general rule. Although the air on aeroplanes is generally dry, it does not seem to increase the risk of catching influenza – perhaps because the air conditioning itself filters out any germs before they have a chance to circulate. And although the dry air seems to fuel the spread of flu in the temperate regions of Europe and North America, some contradictory results suggest the germs may act somewhat differently in more tropical areas.
One explanation is that in particularly warm and wet conditions of a tropical climate, the virus may end up sticking to more surfaces within a room. So although it can’t survive in the air so well, the flu virus could instead be thriving on everything that you touch, making it more likely to pass from hand to mouth.
The BBC also has some more "huh" material:
What’s more, water vapour in the air seems to be toxic to the virus itself. Perhaps by changing the acidity or salt concentration in the packet of mucus, moist air may deform the virus’s surface, meaning that it loses the weaponry that normally allows us to attack our cells. In contrast, viruses in drier air can float around and stay active for hours – until it is inhaled or ingested, and can lodge in the cells in your throat.
(So why is it that that "water vapour toxicity for viruses" does not manifest itself on wet surfaces??)
I'll have to dig up more actual research for a better/real answer, but it seemed a good idea to post this "it depends" answer for now. (I'll accept a better one.)
The BBC does not clearly cite the guinea pigs study they mention, but I suspect it's this 2007 one, by Lowen et al., since it has over 1000 citations
We found that low relative humidities of 20%–35% were most favorable, while transmission was completely blocked at a high relative humidity of 80%. Furthermore, when guinea pigs were kept at 5 °C, transmission occurred with greater frequency than at 20 °C, while at 30 °C, no transmission was detected. Our data implicate low relative humidities produced by indoor heating and cold temperatures as features of winter that favor influenza virus spread.
I'll have to look through what cites it now to see if it's cited because definitely true or cited because it's controversial...
One of the (also pretty cited) papers citing Lowen has this to say:
Person-to-person transmission is central to seasonal and pandemic spread; nevertheless, the modes of spread are a matter of ongoing debate. Resolution of this discussion is paramount to the development of effective control measures in health care and community settings. Using the guinea pig model, we demonstrated that transmission of influenza A/Panama/2007/1999 (H3N2) virus through the air is efficient, compared with spread through contaminated environmental surfaces (fomites). We also examined the aerosol transmission efficiencies of 2 human influenza virus A strains and found that A/Panama/2007/1999 influenza virus transmitted more efficiently than A/Texas/36/1991 (H1N1) virus in our model. The data provide new and much-needed insights into the modes of influenza virus spread and strain-specific differences in the efficiency of transmission.
[...] We have previously established that transmission of human Pan99 is highly efficient by direct contact and at short range through the air in the guinea pig model, despite the absence of expulsion events, such as coughing or sneezing [16–18].
So it appears the guinea pig flu in general transmits poorly on surfaces (fomites), but that might not be the case for all "flus".
That impression appears to be correct based on a 2014 review:
Influenza and respiratory syncytial virus (RSV) are similarly structured viruses with similar environmental survival, but different routes of transmission. While RSV is transmitted predominantly by direct and indirect contact, influenza is also transmitted by aerosol. The cold, dry conditions of temperate winters appear to encourage the transmission of both viruses, by increasing influenza virus survival in aerosols, and increasing influenza and RSV survival on surfaces. In contrast, the hot, wet conditions of tropical rainy seasons appear to discourage aerosol transmission of influenza, by reducing the amount of influenza virus that is aerosolized, and probably also by reducing influenza survival in aerosol. The wet conditions of tropical rainy seasons may, however, encourage contact transmission of both viruses, by increasing the amount of virus that is deposited on surfaces, and by increasing virus survival in droplets on surfaces. This evidence suggests that the increased incidence of influenza and RSV in tropical rainy seasons may be due to increased contact transmission. This hypothesis is consistent with the observation that tropical rainy seasons appear to encourage the transmission of RSV more than influenza. More research is required to examine the environmental survival of respiratory viruses in the high humidity and temperature of the tropics.
It also discusses the complex dynamics of RSV survival in droplets:
One study examined the effect of humidity on RSV
survival in 1 μl droplets of tissue culture medium on
polythene at room temperature. Over the first 5
h, RSV survival was highest at the highest humidity,
while over the next 67 h, RSV survival was highest
at the lowest humidity. The explanation of
these findings may lie in the droplet drying time in
this study. Droplets exposed to 77% RH were still
wet at 18 h (no data were given for drying times at
32% or 52% RH). The relatively high survival at
higher humidity over the first 5 h was probably due
to the fact that the droplets remained wet in these conditions.
The survival over the final 48 h (when all droplets
were dry) was progressively reduced with increasing humidity. Consistent with this explanation,
only 1% of RSV was lost over the 72 h when stored in
liquid culture medium, and in addition, the authors
noted that RSV survival was increased with increased
droplet size. Similarly, in another study the survival of
RSV on countertops was reduced if the virus was in
droplets that were dried quickly. These results
are consistent with the studies examining influenza
survival on surfaces, suggesting that while the virus remains
‘wet’ in droplets, high humidity prolongs its
survival, by reducing evaporation.
Regarding influenza, these graphs from the review should be self-explanatory:
As the (2012) paper from which the 2nd (V-shaped) graph originates explains:
There appear to be three regimes of IAV [influenza A virus] viability in droplets, defined by humidity: physiological conditions (∼100% RH) with high viability, concentrated conditions (50% to near 100% RH) with lower viability depending on the composition of media, and dry conditions (<50% RH) with high viability. This paradigm could help resolve conflicting findings in the literature on the relationship between IAV viability in aerosols and humidity, and results in human mucus could help explain influenza’s seasonality in different regions.
Also quite interesting (but predictable at this point) observation from a different 2015 review:
There had not been any studies of the effect of AH [absolute humidity] on
influenza virus transmission (IVT) until Shaman and Kohn
used the data published by Lowen et al. to recalculate AH
from the RH and temperature values and model its effect
on IVT and influenza virus survival (IVS). Based on the
observation that there a strong seasonal cycle in AH, both
outdoor and indoor, with lowest values during the colder
winter months in temperate regions, they hypothesised
that AH was more likely to affect viral survival and transmission
than RH which is often high in the winter. Using
linear regression to plot temperature, RH and AH against
IVT they demonstrated that AH had a highly significant
inverse relationship (p < 0.00027) with viral transmission;
i.e. the lower the AH, the more viral transmission occurred.
The statistical significance of this was much greater than
for RH and IVT as well as temperature and IVT.
(I'll try to clean up this answer a bit later on.)