It's a common phrase. The people I loosely call "vegetarians" have different rules for what they will and won't eat, and need quick ways to explain them to people. "Vegans" accept no animal products whatever. "Ovolactarians" accept eggs and milk, because they are obtained without killing the animal. "Pescatarians" will eat seafood. My brother said "no red meat," meaning he would eat chicken and fish, but not beef or pork. A friend of mine said "I don't eat mammals," which means about the same thing--the idea being that mammals have well-developed brains and experience physical pain and emotions. "Nothing that has a face" is another codification. I think that includes poultry and fish, but I'm not quite sure. I've also heard "nothing that has a face or a mother." The appeal of "nothing that has a face" is that it isn't cold or technical-sounding and you don't need a dictionary to check the meaning. It also goes part way to explaining the reason--"having a face" indicates some kind of biological kinship.

It's not an expression which is why you can't find it in a dictionary. But it's pretty clear what it means. Animals have faces. The script is talking about 'weekday veg', 'nothing with a face...' Basically, don't eat animals (meat and fish) during the week. At the weekend you can eat what you want.

I'm not familiar with this particular TEDTalk, Steven, but from the context you gave I guess that the presenter talks about not eating meat during the week. 'nothing with a face' = not an animal (as animals have faces, unlike vegetables)

nothing with a face refers to food that aren't animal based :) so basically they're talking about fruit veg anything that doesn't have a face :) hope this helps :)

Interesting that you drew faces on your bananas.