The playground tile installation video shows the tiles ganged together with the corner connector and nothing more, but your website suggests "important tools for installation" including adhesive. I assume that I don't really need adhesive. In what application might you need adhesive?

To adhere to the ramps and corners you may want to use an adhesive (if needed). Just apply the glue to the side that butts up to the tile, press, and hold and the ramp and corners will bind to the floor. Sometimes customers also like to glue their tiles down for a more permanent solution.