Middle English blechen, from Old English blæcan, of cloth or fabric, "to make white by removing color, whiten" (by exposure to chemical agents or the sun), from Proto-Germanic *blaikjan "to make white" (source also of Old Saxon blek, Old Norse bleikr, Dutch bleek, Old High German bleih, German bleich "pale;" Old Norse bleikja, Dutch bleken, German bleichen "to make white, cause to fade"), from PIE root *bhel- (1) "to shine, flash, burn," also "shining white."
The same root probably produced black (q.v.), perhaps because both black and white are colorless, or because both are associated in different ways with burning. For the contrary senses, compare Old English scimian, meaning both "to shine" and "to dim, grow dusky, grow dark," which is related to the source of shine.
The intransitive sense of "become white" is from 1610s. Related: Bleached; bleaching. The past participle in Middle English was sometimes blaught.