Middle English mēte, from Old English mete "food, nourishment, sustenance" (paired with drink), "item of food; animal food, fodder," also "a meal, repast," from Proto-Germanic *mati (source also of Old Frisian mete, Old Saxon meti, Old Norse matr, Old High German maz, Gothic mats "food," Middle Dutch, Dutch metworst, German Mettwurst "type of sausage"), from PIE *mad-i-, from root *mad- "moist, wet," also with reference to food qualities, (source also of Sanskrit medas- "fat" (n.), Old Irish mat "pig;" see mast (n.2)).
Narrower sense of "flesh of warm-blooded animals killed and used as food" is attested from c. 1300 (earlier this was flesh-meat, early 12c.). There is a similar sense evolution in French viande "meat," originally "food." In Middle English, vegetables still could be called grene-mete (15c.) and white meat was "a dairy food or product" (early 15c.). Figurative sense of "essential part" is from 1901.
Dark meat and light meat in reference to the meat of fowls, based on the color when cooked, were popularized 19c., supposedly as euphemisms for leg or thigh and breast, but earliest sources use both sets of terms without apparent embarrassment.
The choicest parts of a turkey are the side bones, the breast, and the thigh bones. The breast and wings are called light meat; the thigh-bones and side-bones dark meat. When a person declines expressing a preference, it is polite to help to both kinds. [Lydia Maria Child, "The American Frugal Housewife," Boston, 1835]
First record of meat loaf is from 1876. Meat-market "place where one looks for sex partners" is from 1896 (meat in various sexual senses of "penis, vagina, body regarded as a sex object, prostitute" are attested from 1590s; Old English for "meat-market" was flæsccyping ('flesh-cheaping')); slang meat wagon "ambulance" is from 1920, American English slang, said to date from World War I (in a literal sense by 1857). Meat-grinder in the figurative sense attested by 1951. Meat-hook in colloquial transferred sense "arm" attested by 1919.