1854, "drinking bout," also (v.) "drink heavily, soak up alcohol;" dialectal use of binge "soak" (a wooden vessel). Said to have been originally as a dialect word. Binge is noted in Evans' "Leicestershire Words, Phrases and Proverbs" (London, 1848) as a dialect verb for "To soak in water a wooden vessel, that would otherwise leak," to make the wood swell. He adds that it was extended locally to excessive drinking ("soaking").
The sense was extended c. World War I to include eating as well as drinking. Binge-watching is from 1996. Related: Binged; bingeing.