Several of the young stellar objects observed with the Hubble Space Telescope in the Orion Nebula near theta(1)C Ori show disklike structures with sizes r similar to 100 AU, similar to our own planetary system. These disklike shapes appear as dark silhouettes in [O III], [S II], [N II], H alpha, and the continuum but are seen in emission in the [O I] lambda 6300 line. We propose in this Letter that the [O I] emission is emerging from a H/H-2 photodissociation front that lies close to the disk surface. The H/H-2 front lies inside a photodissociation region between the disk surface and an ionization front that typically has a standoff distance of several disk radii from the disk surface. OH is produced efficiently at the warm H/H-2 front by the endothermic chemical reaction O + H-2 --> OH + H. However, OH is also efficiently destroyed by photodissociation caused by FUV photons. Approximately 50% of the photodissociated OH produces electronically excited atomic oxygen in the upper level of the 6300 Angstrom transition, which radiatively decays as intense [O I] lambda 6300 emission. Essentially, broadband FUV photons are absorbed by OH and efficiently converted to lambda 6300 line photons. The theoretically predicted [O I] lambda 6300 emission agrees well with that observed in 182-413 (HST -10), the best-studied object with a clearly resolved disk. The H/H-2 photodissociation front is close to the disk surface of 182-413, and the [O I] line, which peaks at the photodissociation front, thus traces the disk surface. The [O I] emission provides additional evidence in a number of proplyds for the existence of an extended PDR between the disk surface and the ionization front, and the penetration of OH-dissociating FUV photons from the ionization front to the disk surface.