I grabbed some video of the last story Mr. S and Ms. T did: a puppet adaptation of That Is Not a Good Idea by Mo Willems.
Before they told the story using the puppets, Mr. S read the book to the group so that they would know how the story goes and when to chime in with their chorus. The little things you can see the kids holding are chick stick puppets. I believe they blew up the chicks from the book and copied them on card stock, then glued them to craft sticks. Before they retold the story with puppets, Mr. S passed out a stick puppet to each child.
Obviously, you can see that they're having a ton of fun! And they're also learning, of course. Getting kids involved in retelling a story is a great way to build narrative skills, one of six early literacy skills that help children get ready to read. Providing the stick puppets encourages kids to talk (one of five practices that helps build early literacy skills), not only when they engage in helping tell the story, but I could hear them continuing to talk about the story as they filed out of the room after the show.
I hope you enjoy the show!