The peregrine falcon Falco peregrinus is renowned for attacking its prey from high altitude in a fast controlled dive called a stoop. Many other raptors employ a similar mode of attack, but the functional benefits of stooping remain obscure. Here we investigate whether, when, and why stooping promotes catch success, using a three-dimensional, agent-based modeling approach to simulate attacks of falcons on aerial prey. We simulate avian flapping and gliding flight using an analytical quasi-steady model of the aerodynamic forces and moments, parametrized by empirical measurements of flight morphology. The model-birds' flight control inputs are commanded by their guidance system, comprising a phenomenological model of its vision, guidance, and control. To intercept its prey, model-falcons use the same guidance law as missiles (pure proportional navigation); this assumption is corroborated by empirical data on peregrine falcons hunting lures. We parametrically vary the falcon's starting position relative to its prey, together with the feedback gain of its guidance loop, under differing assumptions regarding its errors and delay in vision and control, and for three different patterns of prey motion. We find that, when the prey maneuvers erratically, high-altitude stoops increase catch success compared to low-altitude attacks, but only if the falcon's guidance law is appropriately tuned, and only given a high degree of precision in vision and control. Remarkably, the optimal tuning of the guidance law in our simulations coincides closely with what has been observed empirically in peregrines. High-altitude stoops are shown to be beneficial because their high airspeed enables production of higher aerodynamic forces for maneuvering, and facilitates higher roll agility as the wings are tucked, each of which is essential to catching maneuvering prey at realistic response delays.