The multi-state constraint Kalman filter (MSCKF) has been proven to be more efficient than graph optimization for visual-based odometry while with similar accuracy. However, it has not been adequately considered and studied for LiDAR-based odometry. In this paper, we propose a novel tightly-coupled LiDAR-inertial odometry based on the MSCKF framework, named MSC-LIO. An efficient LiDAR same-plane cluster (LSPC) tracking method, without explicit feature extraction, is present for frame-to-frame data associations. The tracked LSPC is used to build an LSPC measurement model that constructs multi-state constraints. Besides, we propose an effective point-velocity-based LiDAR-IMU time-delay (LITD) estimation method, which is derived from the proposed LSPC tracking method. To validate the effectiveness and robustness of the proposed method, we conducted extensive experiments on both public datasets and real-world environments. The results demonstrate that the proposed MSC-LIO yields higher accuracy and efficiency compared to the state-of-the-art methods. Ablation experiments indicate that the data-association efficiency is improved by nearly 3 times with the LSPC tracking, and the proposed LITD estimation method can effectively and accurately estimate the LITD. Besides, MSC-LIO was implemented on an edge device and demonstrated excellent real-time performance.