Humans race drones faster than algorithms, despite being limited to a fixed camera angle, body rate control, and response latencies in the order of hundreds of milliseconds. A better understanding of the ability of human pilots of selecting appropriate motor commands from highly dynamic visual information may provide key insights for solving current challenges in vision-based autonomous navigation. This paper investigates the relationship between human eye movements, control behavior, and flight performance in a drone racing task. We collected a multimodal dataset from 21 experienced drone pilots using a highly realistic drone racing simulator, also used to recruit professional pilots. Our results show task-specific improvements in drone racing performance over time. In particular, we found that eye gaze tracks future waypoints (i.e., gates), with first fixations occurring on average 1.5 seconds and 16 meters before reaching the gate. Moreover, human pilots consistently looked at the inside of the future flight path for lateral (i.e., left and right turns) and vertical maneuvers (i.e., ascending and descending). Finally, we found a strong correlation between pilots eye movements and the commanded direction of quadrotor flight, with an average visual-motor response latency of 220 ms. These results highlight the importance of coordinated eye movements in human-piloted drone racing. We make our dataset publicly available.