Large-amplitude oscillations of foils have been observed to yield greater propulsive efficiency than small-amplitude oscillations. Using scaling relations and experiments on foils with peak-to-peak trailing edge amplitudes of up to two chord lengths, we explain why this is so. In the process, we reveal the importance of drag, specifically how it can significantly reduce the efficiency, and how this effect depends on amplitude. The scaling relations and experimental data also reveal a fundamental tradeoff between high thrust and high efficiency, where the drag also plays a crucial role.