(noun) "animal flesh as a food"
This is a Germanic word known from Anglo-Saxon times, where it meant "food in general" - a usage still occasionally heard, for example in the phrase meat and drink ("food and drink"). The modern meaning developed in the 14th century. Both old and modern senses are thus to be found in Shakespeare, and it is not always clear which is required.
But there are several cases where the older meaning is clear. When Jack Cade says "I have eat no meat these five days" (Henry VI Part 2, IV.x.37) he is starving for lack of any food, not just fleshmeat, and the same applies to Katharina when she says she is "starved for meat" (The Taming of the Shrew, IV.iii.9).
The old sense of "nourishment" is even clearer when Mercutio tells Benvolio "Thy head is as full of quarrels as an egg is full of meat" (Romeo and Juliet, III.i.22), and the word means "edible part" when the Fool tells Lear he has "cut the egg i'the middle and eat up the meat" (King Lear, I.iv.157).