How does this learning activity develop this particular skill?

The Jigsaw method structurally requires students to connect with and pay attention to one another. A sense of positive interdependence is developed in both the expert group and the jigsaw group. Through this, students come to understand that they cannot be as successful on their own as they can as a group, and that they need their peers. 

In both cases (mosaic group, expert group), a heterogeneous team is formed. In the expert group, even the least academically gifted child must understand the given lesson content, so group members learn to provide responsible assistance to those in need, as well as that anyone can have a good idea regarding a given topic.

When an anxious, marginalized, or less skilled student returns to their group as an “expert,” the others are compelled to pay attention to them and value their knowledge, which radically increases their status and their connection to the group.