The article is really short, so it's not really possible to summarize it. Both authors are primatologists, and they see things in a perspective that is quite different from the one often taken by those whose interest in primate social cognition is mainly (if not only) driven by an interest in the evolution of human cognition (I clearly fit in this category, so I’m not blaming anybody). When they list the selection pressures that might have driven the evolution of social intelligence, they don’t stop (as one can be tempted to do) at cooperation and competition. They also mention dealing with predation and enlarging the diet, and they give evidence that both of these domains could have driven the evolution of some sort of social intelligence.

When they speak of the second domain (enlarging the diet), they hint at a funny hypothesis: that “human capacities to understand mental states derive ultimately from selection pressures on non-human apes to learn complex feeding skills”. I clearly don’t know enough about ape and early hominids feeding behaviour to judge its plausibility, but it dovetails nicely with the ideas on the importance of pedagogy put forward by Csibra and Gergely (see this post) on the one hand, and with the idea I’ve had for some time that the evolution of mindreading abilities can only had been driven, in the beginnings, by some form of cooperation (see this poster we did with Nicolas Baumard). Moreover, since this kind of teaching can be directed mostly to the apes’ own children, it is possible to envision a bootstrapping with kin selection.