Low Earth Orbit (LEO) satellite communication presents a promising solution for delivering Internet access to users in remote regions. Given that video content is expected to dominate network traffic in LEO satellite systems, this study presents a new video-aware mobility management framework specifically designed for such networks. By combining simulation models with real-world datasets, we highlight the critical role of handoff strategies and throughput prediction algorithms in both single-user and multi-user video streaming scenarios. Building on these insights, we introduce a suite of innovative algorithms that jointly determine satellite selection and video bitrate to enhance users' quality of experience (QoE). Initially, we design model predictive control (MPC) and reinforcement learning (RL) based methods for individual users, then extend the approach to manage multiple users sharing a satellite. Notably, we incorporate centralized training with distributed inference in our RL design to develop distributed policies informed by a global view. The effectiveness of our approach is validated through trace-driven simulations and testbed experiments.