"become putrid," hence "be spoiled, be made worthless or ineffective," 1640s (implied in addled), from archaic addle (n.) "urine, liquid filth," from Old English adela "mud, mire, liquid manure" (cognate with East Frisian adel "dung," Old Swedish adel "urine," Middle Low German adel "mud," Dutch aal "puddle").
Popularly used in the noun phrase addle egg (mid-13c.) "egg that does not hatch, rotten egg," a loan-translation of Latin ovum urinum, literally "urine egg," which is itself an erroneous loan-translation of Greek ourion ōon "putrid egg," literally "wind egg," from ourios "of the wind" (confused by Roman writers with ourios "of urine," from ouron "urine").
From this phrase, since c. 1600 the noun in English was mistaken as an adjective meaning "putrid," and thence given a figurative extension to "empty, vain, idle," also "confused, muddled, unsound" (1706), then back-formed into a verb in that sense. Related: Addling.
Popular in forming derogatory compounds 17c. and after, such as addle-headed "stupid, muddled" (1660s); addle-pated (1630s); addle-pate "stupid bungler" (c. 1600); addle-plot "spoil-sport, person who spoils any amusement" (1690s).