The structure is a genuine anatomical oddity. The posterior thoracic ribs are greatly elongated and hinged so they can rotate outward from the body wall, tenting a flap of skin into an aerofoil. At rest the ribs fold back and the membrane collapses against the flanks, leaving a lizard that looks unremarkable on a trunk.
In flight the animal launches from a trunk, opens the patagium, and glides in a shallow descent, typically covering something on the order of several metres up to around thirty in reported cases, with a small neck flap acting in stabilisation. Filming has shown that the forelimbs grip the leading edge of the membrane and adjust its shape, so the lizard is controlling the aerofoil rather than merely falling with a parachute.
This is gliding, not powered flight; there is no flapping and no way to gain altitude. Whether the primary selective driver was escape from predators, efficient travel between trunks, or access to territory has not been settled, and the different Draco species vary enough in size and membrane loading that one answer may not fit all.