This paper delves into an integrated sensing and communication (ISAC) system bolstered by a simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS). Within this system, a base station (BS) is equipped with communication and radar capabilities, enabling it to communicate with ground terminals (GTs) and concurrently probe for echo signals from a target of interest. Moreover, to manage interference and improve communication quality, the rate splitting multiple access (RSMA) scheme is incorporated into the system. The signal-to-interference-plus-noise ratio (SINR) of the received sensing echo signals is a measure of sensing performance. We formulate a joint optimization problem of common rates, transmit beamforming at the BS, and passive beamforming vectors of the STAR-RIS. The objective is to maximize sensing SINR while guaranteeing the communication rate requirements for each GT. We present an iterative algorithm to address the non-convex problem by invoking Dinkelbach's transform, semidefinite relaxation (SDR), majorization-minimization, and sequential rank-one constraint relaxation (SROCR) theories. Simulation results manifest that the performance of the studied ISAC network enhanced by the STAR-RIS and RSMA surpasses other benchmarks considerably. The results evidently indicate the superior performance improvement of the ISAC system with the proposed RSMA-based transmission strategy design and the dynamic optimization of both transmission and reflection beamforming at STAR-RIS.