The gizmo is to control and activate the actual brakes on trailer wheels (presuming the trailer has electric brakes). The brake lights are separate, and no special controller is generally needed for them; they're just switched on with the vehicle brake lights.
(Technically, trailers do use combined brake and turn lights, so there may be a sort of control circuit needed if the tow vehicle has separate brake and turn lights. You wouldn't see such a light circuit adapter box accessible on the dashboard, nor does it have any adjustments or controls to operate.)