Whisker stress is commonly believed to be discomfort caused by the cat's (very sensitive) whiskers brushing against the sides of a food bowl. Ingrid King (an author who previously spent 10+ years working in a veterinary office) describes it:
Whiskers are extremely sensitive, and when a food bowl is too narrow and too deep, a cat is forced to put her face all the way into the bowl to reach her food. This causes her whiskers to bump against the side of the bowl, which causes discomfort. In extreme cases, cats may refuse to eat out of deep, narrow bowls altogether.
The pictures (not drawings) of the meme mentioned by the question tend to have very steep sides, so the cat will eat out of the center (where it's comfortable and the whiskers have plenty of room), but decline to eat the food on the sides (where the whiskers would rub against one side of the bowl).
I'm not entirely convinced about this.
I tried to find any research to indicate that whisker stress is a real phenomenon for an earlier question and could not find any scholarly (peer reviewed) articles about it. I just found multiple instances of cat owners telling other cat owners about it (and selling special bowls!).
Additionally, we give a bit of dry food in the Flower Slo-Bowl and sometimes the cats ask for fresh even when there is still food in the bowl. They do not eat directly out of the bowl, but have to spend some time "fishing" kibble out of the bowl and onto the floor, where they then eat it (we do this before bed so they are entertained while we sleep). They may just be asking for food that they don't have to work for (wet food meals are fed on platters), so it's hard to tell if this disproves the theory or not.