What do battens do on a sail?

Answer 1

Battens help give the sail it's shape.

By adding these 'spars' to the sail they control how the sail deforms under loading, resulting in a sail shaped more like a wing and less like a parachute.

Battens also allow for a larger sail area, assuming the boom length and mast height remain constant. With a traditional non-battened sail, the largest possible functioning sail is a triangle defined by the ends of the boom and the tip of the mast. With battens, the sail can extend past that line drawn from the back of the boom to the tip of the mast, as the battens will act like cantilevered beams, supporting the sail past the point where tension between the boom and mast can be used. As a result, all non-battened sails are triangles, and almost every decent sail has an arched trailing edge.