A trait is said to be adaptive when it causes fitness to increase. Fitness is generally understood as the (relative) contribution to future generations in terms of offspring or genes. The trait is selected for by the environment and hence increases fitness.
In the paper of Dey et al. this is the fitness of the parent birds. Hatching asynchrony causes size differences between chicks which may (1) "facilitate brood reduction in unfavourable environmental conditions", (2) "provide insurance against the failure of core offspring" and (3) "reduce competition among nestlings". Dey et al. also cite a reason why hatching asynchrony may be sustained in a population despite being maladaptive: initiating incubation before the clutch is completely laid is selected for and causes asynchronous hatching as a by-product. In this case, hatching asynchrony merely correlates with fitness, but does not cause fitness to increase and is thus not adaptive.