This paper considers reconfigurable intelligent surface (RIS)-enabled federated learning (FL) system, where the FL users communicate with the access point (AP) via RIS. To reveal the impact of RIS and learning rate on FL aggregation, the theoretical result of minimum global communication rounds and local iteration rounds are derived. Based on the obtained convergence results of FL, we formulate an optimization problem to minimize the energy consumption of the proposed RIS-assisted FL system by jointly optimizing the passive beamforming of RIS, the CPU computing frequency, the bandwidth, and the transmit power of users. To solve the non-convex problem, we propose a block coordinate descent (BCD) optimization algorithm based on successive convex approximation (SCA) to decompose the original problem into four sub-problems. Specifically, the closed-form solutions are derived for the CPU frequency, RIS reflection matrix, and communication bandwidth. For the transmit power sub-problem, we propose a linear approximation algorithm based on the first-order Taylor expansion to ensure solution accuracy. Finally, simulation results show that: 1) the energy consumption of the proposed RIS-assisted FL system can be greatly reduced compared to that without optimizing the passive beamforming of RIS and the transmit power; 2) The learning performance of the proposed RIS-enabled FL system is closed to the FL without wireless communication interference; and 3) The proposed algorithm can not only significantly reduce energy consumption, but also fast convergence in terms of the FL model training and testing.