Meat == food, not just flesh. This is a very old definition for the word. It's also still used to describe the bits of nuts that we eat ("coconut meat").
meat, n.
1. a. Food in general; anything used as nourishment for men or animals; usually, solid food, in contradistinction to drink. Now arch. and dial. green meat: grass or green vegetables used for food or fodder (see green a. 4). See also hard meat, horsemeat, whitemeat. meal of meat, meal's meat: see meal n. 2 1e.
a900: tr. Bæda's Hist. v. iv. (Schipper) 568 “He eode on his hus & þær mete [v.r. mæte] þyᵹede.”
c975: Rushw. Gosp. Luke xii. 23 “Sawel mara is ðonne mett.”
a1050: Liber Scintill. xlvii. (1889) 153 “Nys rice godes meta & drinc.”
c1175: Lamb. Hom. 135 “Ne sculen ȝe nawiht ȝimstones leggen Swinen to mete.”
c1200: Ormin 3213 “Hiss drinnch wass waterr aȝȝ occ aȝȝ, Hiss mete wilde rotess.”
a1240: Lofsong in Cott. Hom. 205 “Ich habbe i-suneged ine mete and ine drunche.”
a1300: Cursor M. 898 “Mold sal be þi mete for nede.”
c1380: Wyclif Wks. (1880) 206 “Alas, þat so gret cost & bisynesse is sette abouten þe roten body, þat is wormes mete.”
c1440: Pol. Rel. & L. Poems (1903) 185 “Thy mete shall be mylk, honye, & wyne.”
1477: Norton Ord. Alch. v. in Ashm. (1652) 76 “Without Liquor no Meate is good.”
1578: Lyte Dodoens ii. xlvi. 205 “These kindes of lillies are neither used in meate nor medicine.”
1623: Cockeram ii, “Meate of the Gods, Ambrosia, Manna.”
meat, n.
e. The edible part of fruits, nuts, eggs, etc.: the pulp, kernel, yolk and white, etc. in contradistinction to the rind, peel, or shell. ?Now only U.S. exc. in proverbial phrase (see quot. 1592). Also, the animal substance of a shell-fish.
c1420: Pallad. on Husb. iii. 708 “A stanry pere is seyd to chaunge his mete In esy lond ygraffed yf he be.”
1530: Palsgr. 245/1 “Meate of any frute, le bon.”
a1562: G. Cavendish Wolsey (1893) 30 “A very fayer orrynge wherof the mete or substaunce within was taken owt.”
1592: Shakes. Rom. & Jul. iii. i. 25 “Thy head is as full of quarrels, as an egge is full of meat.”
1613: Purchas Pilgrimage (1614) 506 “Of the meat of the Nut dried, they make oyle.”
1679: J. Skeat Art Cookery 30 “First take all the meat out of the lobster.”
1766: Museum Rust. I. lxxxiii. 370 “Low or swampy grounds don't answer well for potatoes,..the meat being generally scabby, close, wet and heavy.”
1802: Paley Nat. Theol. xx. (1819) 313 note, “The meat of a plum.”
1900: Boston Even. Transcr. 29 Mar. 7/3 “Force through a meat chopper with one-half pound nut-meats, using English walnut meats, pecan-nut meats.”
1902: Fortn. Rev. June 1012 “A bit of crab-meat.”