LiDAR-based 3D perception and localization on unmanned aerial vehicles (UAVs) are fundamentally limited by the narrow field of view (FoV) of compact LiDAR sensors and the payload constraints that preclude multi-sensor configurations. Traditional motorized scanning systems with fixed-speed rotations lack scene awareness and task-level adaptability, leading to degraded odometry and mapping performance in complex, occluded environments. Inspired by the active sensing behavior of owls, we propose AEOS (Active Environment-aware Optimal Scanning), a biologically inspired and computationally efficient framework for adaptive LiDAR control in UAV-based LiDAR-Inertial Odometry (LIO). AEOS combines model predictive control (MPC) and reinforcement learning (RL) in a hybrid architecture: an analytical uncertainty model predicts future pose observability for exploitation, while a lightweight neural network learns an implicit cost map from panoramic depth representations to guide exploration. To support scalable training and generalization, we develop a point cloud-based simulation environment with real-world LiDAR maps across diverse scenes, enabling sim-to-real transfer. Extensive experiments in both simulation and real-world environments demonstrate that AEOS significantly improves odometry accuracy compared to fixed-rate, optimization-only, and fully learned baselines, while maintaining real-time performance under onboard computational constraints. The project page can be found at https://kafeiyin00.github.io/AEOS/.