It’s less than an hour before the Dave Matthews Band takes the stage on a sunny Thursday evening on the coast of Long Island—but the biggest crowds at the Northwell at Jones Beach Theater aren’t at the tequila bar. They’re in the “eco-village” operated by Reverb, a nonprofit focused on greening live music by inspiring fans to take action around climate change.

As I wander through tents emblazoned with the logos of organizations like the Nature Conservancy and Generation180, volunteers explain how...