The flexible arresting cable in the recovery cable system is controlled to capture the landing rocket and achieve safe recovery. The high-dimensional nonlinear dynamic problems of the flexible cable urgently request elegant simplifications and model reductions for use in the real-time state estimation and control applications. In this work, a da-ta-driven reduced-order modeling approach is proposed based on the Proper Orthogonal Decomposition Galerkin (POD-Galerkin) projection and the Sparse Identification of Nonlinear Dynamical Systems (SINDy) algorithm. The accurate nonlinear flexible multibody dynamic model of the arresting cable is established using Arbitrary Lagrangian Eulerian (ALE) formulation. Then, it is simulated to generate a comprehensive data matrix. The POD-Galerkin projection carries out an approximate representation of the higher-order data matrix through a reduced set of orthogonal basis functions. Subsequently, benefiting by compressed sensing and sparse regression techniques, the reduced-order model of the flexible cable can be identified utilizing the SINDy algorithm and can be represented through the nonlinear candidate function library. The simulation results show that the dynamic model of a flexible cable effectively reduces the order using POD-Galerkin projection from 45° of freedom to only four degrees of freedom, and the dynamic behavior of the identified model maintaining excellent agreement with the dynamic behavior of the original model.