Generally it means eating more plants, less sugar, refined or otherwise, avoiding gluten, cutting down on carbs, and of course cutting out red meat, salt and laughing. I made the last one up. Or maybe I didn’t. As LV Anderson put it on Slate.com: “Clean eating can mean pretty much anything you want it to mean.”
Mostly what it means is: “I’m much better than you.” The opposite of clean is dirty. There’s dirty politics, dirty money and dirty dealings. People who enjoy sex are portrayed as dirty. (Though Woody Allen’s line that sex was only dirty “when it was done properly” is instructive here.) When junkies kick the habit they “get clean”. Bad people “clean up their act”. In short, if you don’t eat clean you are lacking in virtue. You are not a good person. You are a bad person. You filthy, dirty dog.