I know that Haferflocken is oats. But what does kernig mean? Leo says that it means meaty or lusty.
What does kernig means in this context?
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Wikipedia has a nice article describing Haferflocken. There you can find an image comparing kernige Haferflocken to zarte Haferflocken.
Basically, kernig means hard or solid, as opposed to the other types of Haferflocken.
After the oat grains (groats) are dehusked, heated and dryed they are then processed into oatmeal - "kernig(e)" Haferflocken are steamed and rolled whole oat groats (old fashioned oatmeal). The word "kernig" is a colloquial term for hard, sturdy, rough, grainy, wholemeal. Kern=Grain (kern-ig = grain-y)
Simply there are two types of Haferflocken on the market:
The difference is, from the consumer's perspective, just the size and structure of the flakes, kernig being the large, coarse ones, fein being the tiny, soft ones. (For differences in the production process, see Diana's answer.)
However, be aware that this is pure commercial terminology. There are various expressions around that are used only by marketing people, or by those who create the packaging of a product, and kernig is clearly such a word. Nobody in everyday life would say kernige Haferflocken. Usually you would say grobe Haferflocken. (Unless you are already brainwashed by commerce-speak.)
Another expression that is used exclusively by marketeers is "Joghurt mild". First, the normal expression would be "milder Joghurt" (adjective first), second, it has become something like a mania to market yoghourt as "mild". You literally will not find packaged yoghourt in Germany that does not claim to be "mild" (speaking about plain yoghourt here, not the varietes with lots of sugar, food colour and usually negligible traces of fruit, marketed as "Fruchtjoghurt"). And as literally every plain yoghourt is marketed as "mild", you could leave the word away as it does not make a difference.
Anyway I always scan the shelves if finally I find a yoghourt marketed as "definitely not mild" and I would buy it right away because good real yoghourt is anything but "mild".