Light-emitting diodes (LEDs) with high efficiencies can be fabricated by a combination of surface texturing and the application of a rear reflector. We demonstrate an external quantum efficiency of 43% for unencapsulated surface-textured thin-film LEDs, which increases to 54% after encapsulation. At low temperatures, the efficiency of unencapsulated devices increases up to 68%. We investigate the light extraction mechanism from such LEDs employing a Monte Carlo simulation of the light propagation inside the LED structure. One essential input parameter for the simulation are the light scattering properties of the textured surface, which have been investigated experimentally. For light incidence below the critical angle of total internal reflection, the transmission through a textured surface is reduced compared to a flat surface. However, due to surface texturing, transmission becomes possible for incident angles above the critical angle. As a result, the internal scattering during internal reflection at the textured surface is not necessary for an efficient extraction of the light generated inside the LED structure. In addition, the Monte Carlo simulation also explains the strong increase of the LED efficiency at low temperatures quantitatively by photon recycling effects. Photon recycling is also demonstrated to be partially responsible for the shift of the emission wavelength in thin- film LEDs, as compared to conventional LEDs.