First of all, this is a paper about internal variability of the climate system in the last decade, not on additional factors that drive climate. Second, this is a discussion about stratospheric water vapour (10 to 15 km above the surface), not water vapour in general. Stratospheric water vapour comes from two sources the uplift of tropospheric water through the very cold tropical tropopause (both as vapour and as condensate), and the oxidation of methane in the upper stratosphere (CH4+2O2 > CO2 + 2H2O NB: this is just a schematic, the actual chemical pathways are more complicated). There isnt very much of it (between 3 and 6 ppmv), and so small changes (~0.5 ppmv) are noticeable.
The decreases seen in this study are in the lower stratosphere and are likely dominated by a change in the flux of water through the tropopause. A change in stratospheric water vapour because of the increase in methane over the industrial period would be a forcing of the climate (and is one of the indirect effects of methane we discussed last year), but a change in the tropopause flux is a response to other factors in the climate system. These might include El Nino/La Nina events, increases in Asian aerosols, or solar impacts on near-tropopause ozone but this is not addressed in the paper and will take a little more work to figure out.
Update: This last paragraph was probably not as clear as it should be. If the lower stratospheric water vapour (LSWV) is relaxing back to some norm after the 1997/1998 El Nino, then what we are seeing would be internal variability in the system which might have some implications for feedbacks to increasing GHGs, and my estimate of that would be that this would be an amplifying feedback (warmer SSTs leading to more LSWV). If we are seeing changes to the tropopause temperatures as an indirect impact from increased Asian aerosol emissions or solar-driven ozone changes, then this might be better thought of as impacting the efficacy of those forcings rather than implying some sensitivity change.
The study includes an estimate of the effect of the observed stratospheric water decadal decrease by calculating the radiation flux with and without the change, and comparing this to the increase in CO2 forcing over the same period. This implicitly assumes that the change can be regarded as a forcing. However, whether that is an appropriate calculation or not needs some careful consideration. Finally, no-one has yet looked at whether climate models (which have plenty of decadal variability too) have phenomena that resemble these observations that might provide some insight into the causes.