German cockroaches carry organisms that cause disease, principally bacteria, protozoans, and viruses that cause gut symptoms. They also produce malodorous secretions that taint the flavor of food, and their cast skins are allergens. All of these issues are enough, but these bugs are also the most aesthetically-unpleasing of pests.
The roach's flat bodies allow them to live in cracks and crevices of human habitations, often in large numbers. German cockroaches scavenge on any food as well as non-food stuffs, including soap and glue. That is why they are commonly found in garbage receptacles.