To "Eat the Meaties" is an affectionate form of vernacular around eating your ladies tender parts (meat curtains) after she's had an exceptionally long absence of such a release. It is rarely met with scorn or disdain as it represent a perfect opportunity for both a sexual act and simultaneous release of pent up frustration. A synonym for cunnilingus and a playful pun on the cereal tagline "Eat your Wheaties." It's clever and women know precisely what it means.
"I'm going to eat the meaties this weekend."
"I wish he we would step up and eat the meaties."
"Better eat you meaties."