Why appliances specifically, and not just the kitchen generally
German cockroaches are thigmotactic, meaning they actively seek out cracks and gaps that touch their bodies on multiple sides at once rather than open, exposed surfaces. A motor housing, a control panel void, or the gap behind a refrigerator's compressor delivers exactly that kind of tight, dark contact. Combine that preference with a hard biological constraint — Purdue research found German cockroach males cannot survive more than about three days without access to water — and appliance interiors become close to ideal shelter: warm around the clock, supplied with condensation from drip pans and seals, and rarely disturbed.
Food debris and grease film finish the picture. Grease accumulates on and around cooking appliances as a matter of course, and it's a genuine food source roaches don't have to travel far to reach. Once a few roaches establish themselves in a spot like this, their droppings release aggregation pheromone that actively signals "safe harborage" to other roaches nearby — which is why an appliance infestation tends to grow in place rather than disperse.

