What exactly does adaptive mean?
The question is probably more complicated than it seems because, if I am not wrong, the word adaptation here is understood at the group level.

Definitions of adaptation

Unfortunately, there is not such thing as a single, standard definition of adaptation. But for most cases the accurate definition the author is using is not of much importance as all of the usual definitions totally fit in the sentence without changing the meaning of the concept they want to express. In you case however, the concept of adaptation seems to be a bit more complicated (and interesting) because it refers to a group-level adaptation.

Note that most people does not really care about the exact definition they're using and this might yield to some confusion. It will probably be a bit hard to know exactly what the authors  (Dey et al.) of your article had in mind.

Definition of Group-level adaptation

I am here going to talk about how the term adaptation can be understood for a (social) group. This is important as I suspect the authors to use the term adaptation on a group phenotypic trait (emergent trait if you want).

Pareto optimality

To my experience, the concept of group-level adaptation is defined in relation to the concept of Pareto optimality which is a concept used in evolutionary game theory. Shortly speaking, in a system that is at Pareto optimality no individual can by any change of the trait under consideration increase its fitness without decreasing the fitness of another individual. In other words, a population that is at Pareto optimality has the greatest achievable population mean fitness. Therefore, it makes sense to consider a group to be (perfectly) adapted when it is at Pareto optimality.

Nash equilibrium

Depending on the underlying evolutionary game, the Pareto optimality may or may not be a Nash equilibrium. Shortly speaking, a population is at Nash equilibrium if no individual can increase its fitness by changing the trait under consideration. Therefore, group-level adaptation can only be stable if the Nash equilibrium is Pareto optimal. An example of game where the Nash equilibrium is the free-market game.

In your case

At first sight, because asynchrony is a group level trait, it seems to me that the sentence "Hatching asynchrony is thought to be adaptive [..]" means that hatching asynchrony is thought to be at Pareto optimality. In other words, it means that hatching asynchrony is thought to be the state of the system where the mean fitness of the group is maximized, that is no individual can increase its fitness without decreasing the fitness of another individual by being synchronous.