Two-fold cost of sex and r/K selection theory
r/K selection has become more of an heuristic for characterizing a species' strategy of reproduction and less a matter of predicting what sort of species will emerge from a given environment as it had originally been proposed by MacArthur and Wilson in 1967. The paper you cite tries to determine whether rotifers become more r or more K selective in different environments. 

The thought here on k-selection is not almost from the definitions of the term, which is why it isn't explained in the paper.  

In R-selected strategies, many more offspring are produced, but less investment is made in each one.  Think about a cockaroach laying thousands of tiny eggs.  The hatchlings might be eaten by their own mother; they are so 'cheap' the insect who layed the eggs doesn't have any anxiety about their survival.  In extreme cases, most of these hatchlings will not reproduce, they may even be a bit of food for the ones which do reach maturity. 

In k-selected strategies, very few offspring are produced.  In addition to the 'cost of sex' limitation, they are invested in so heavily that the female may stop being fertile for a while.  An example would be how female mammals which are lactating are not fertile or seasonal mating like many birds do; the females turn off reproduction so that they can focus on the success of one or two produced offspring. 

k-selection is advantageous over R-selection when more of their investment pays off.  If one out of ten R-offspring survive and one out of four k-selected offspring survive, then for these competing rotifers, the k-selected strategy is advantageous. 

The cost of sex is that its half as efficient as asexual reproduction because males don't really reproduce - they only provide genetic material for females.  You can argue r/K selection independently of this factor, but the article is suggesting that k-selection can be so advantageous as to also give sexual selection an advantage over asexual selection.  I think asexual reproduction could be k-selected as well, but that seems to be the point being made here. 

The reason that k/R has toned down in importance in the past couple of decades is that it has been found that individual species can adapt to shift their k/R strategems.