During their final transformation, insects emerge from the pupal case and deploy their wings within minutes. The wings deploy from a compact origami structure, to form a planar, rigid and functional blade that allows the insect to fly. The deployment is powered by a rapid increase in internal pressure, and by the subsequent flow of hemolymph into the deployable wing structure. Using a combination of imaging techniques, we characterize the internal and external structure of the wing in Drosophila melanogaster , the unfolding kinematics at the organ scale, and the hemolymph flow during deployment. We find that beyond the mere unfolding of the macroscopic folds, wing deployment also involves an expansion of cell surface and the unfolding of microscopic wrinkles in the cuticle enveloping the wing. A quantitative computational model, incorporating mechanical measurements of the viscoelastic properties and microstructure of the wing, predicts the existence of an operating point for deployment and captures the dynamics of expansion. This model suggests that insects exploit material and geometric nonlinearities to achieve rapid and efficient reconfiguration of soft deployable structures.