1520s, "to render (a hunting animal) eager for prey by rewarding it with flesh from a kill," with figurative extensions, from flesh (n.). Meaning "to clothe or embody with flesh," with figurative extensions, is from 1660s.
1530s, "persons united in a body for some purpose," from such use in Anglo-Latin, from L. corporationem, noun of action from corporare "to embody" (see corporate). Meaning "legally authorized entity" (including municipal governments and modern business companies) is from 1610s.