Legged robot control has improved in recent years with the rise of deep reinforcement learning, however, much of the underlying neural mechanisms remain difficult to interpret, especially those that drive robust behavior. My work addresses this gap, by leveraging bio-inspired methods from computational neuroscience to better understand the high-dimensional neural activity of robot locomotion controllers. First, I find that embodied robotic agents exhibit smooth dynamics that reduce tangling -- or opposing neural trajectories in neighboring neural space -- a core principle of recurrently-driven systems, such as the primate motor cortex. Second, I find that recurrent state dynamics are structured and low-dimensional, and that unforced dynamics are governed by fixed point topologies that become increasingly complex with training time. Lastly, I analyze the forced dynamics by pairing physical disturbance trials with various neural ablation strategies. Data from these trials provide evidence that lateral stability is driven by both sensory feedback and recurrent neural activity. Through biomechanical and neural recordings, I find that recovery behavior is characterized by agile hip actuation that drives the front and rear outer legs to swing out and stabilize the agent. This result has been found to be generalizable, as it is seen across various robot embodiments. This framework combines model-based and sampling-based ablation methods for drawing causal relationships between recurrent neural network activity and robust embodied robot behavior.