Here is an example of an item with no dairy ingredients that is milchig. Take an egg. Boil it in milk. Peel the egg. The egg is milchig, but contains no dairy ingredients.
Similarly with meat. Deep fry chicken in oil. Use the same oil to deep fry French fries. The French fries are fleishig, and require you to wait 6 hours after eating them, but contain no meat.
The question is not whether it contains milk or meat, but rather whether the taste of milk or meat has been transferred to it.