A simultaneously transmitting and reflecting surface (STARS) enabled edge caching system is proposed for reducing backhaul traffic and ensuring the quality of service. A novel Caching-at-STARS structure, where a dedicated smart controller and cache memory are installed at the STARS, is proposed to satisfy user demands with fewer hops and desired channel conditions. Then, a joint caching replacement and information-centric hybrid beamforming optimization problem is formulated for minimizing the network power consumption. As long-term decision processes, the optimization problems based on independent and coupled phase-shift models of Caching-at-STARS contain both continuous and discrete decision variables, and are suitable for solving with deep reinforcement learning (DRL) algorithm. For the independent phase-shift Caching-at-STARS model, we develop a frequency-aware based twin delayed deep deterministic policy gradient (FA-TD3) algorithm that leverages user historical request information to serialize high-dimensional caching replacement decision variables. For the coupled phase-shift Caching-at-STARS model, we conceive a cooperative TD3 \& deep-Q network (TD3-DQN) algorithm comprised of FA-TD3 and DQN agents to decide on continuous and discrete variables respectively by observing the network external and internal environment. The numerical results demonstrate that: 1) The Caching-at-STARS-enabled edge caching system has advantages over traditional edge caching, especially in scenarios where Zipf skewness factors or cache capacity is large; 2) Caching-at-STARS outperforms the RIS-assisted edge caching systems; 3) The proposed FA-TD3 and cooperative TD3-DQN algorithms are superior in reducing network power consumption than conventional TD3.