I found an answer to this mystery in the literature, but it is a somewhat tentative explanation from a relatively recent (2012) paper by Yang et al. In a nutshell, [respiratory] viruses do worse in small droplets compared to large/unconstrained ones. Even being "totally" dry is apparently somewhat better for them than being trapped in a small droplet. So you get a V (or U)-shaped survival curve as a function of relative humidity.
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.
After release from the respiratory tract, where RH is ∼100%, a respiratory droplet shrinks by 40–50% in diameter at RH below 90% due to evaporation –. As a result, concentrations of solutes in the droplet increase by up to 15 times, and solutes such as salts (e.g., sodium chloride (NaCl)) that are harmless at physiological levels may become harmful to the virus. For example, avian IAV has been reported to be less stable at salinities greater than 25 g L−1 . Evaporation induces changes to IAV’s microenvironment inside droplets that may affect the virus’ viability, and the toxic effect of solutes may be enhanced at lower RH due to higher concentrations that result from greater loss of water.
Yang et al.'s paper has been positively cited in a couple of subsequent reviews Lowen and Steel (2014) and Paynter (2014)
So it seems to be a fairly credible explanation as far as influenza is concerned. Paynter's review also discusses similar observations on RSV:
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.
There might be more recent advances in this area though, especially regarding mechanisms of the "middle regime".