Non-volatile memories-providing the information storage functionality-are crucial circuit components. Solution-processed organic ferroelectric memory diodes are the non-volatile memory candidate for flexible electronics, as witnessed by the industrial demonstration of a 1 kbit reconfigurable memory fabricated on a plastic foil. Further progress, however, is limited owing to the lack of understanding of the device physics, which is required for the technological implementation of high-density arrays. Here we show that ferroelectric diodes operate as vertical field-effect transistors at the pinch-off. The tunnelling injection and charge accumulation are the fundamental mechanisms governing the device operation. Surprisingly, thermionic emission can be disregarded and the on-state current is not space charge limited. The proposed model explains and unifies a wide range of experiments, provides important design rules for the implementation of organic ferroelectric memory diodes and predicts an ultimate theoretical array density of up to 1012 bit cm-2.