Of small extent across. Thinness lets light and pressure through that thickness would stop.
Old English þynne (thin, slender), from Proto-Germanic *thinniz, ultimately a Proto-Indo-European root for stretching; a thing made thin has, since the deepest root, been stretched rather than simply reduced.
Narrow enough across that what a thicker barrier would block gets through anyway — light, pressure, a signal. Slightness here is a permission, not merely a measurement.