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Federated learning (FL), as a disruptive machine learning paradigm, enables the collaborative training of a global model over decentralized local datasets without sharing them. It spans a wide scope of applications from Internet-of-Things (IoT) to biomedical engineering and drug discovery. To support low-latency and high-privacy FL over wireless networks, in this paper, we propose a reconfigurable intelligent surface (RIS) empowered over-the-air FL system to alleviate the dilemma between learning accuracy and privacy. This is achieved by simultaneously exploiting the channel propagation reconfigurability with RIS for boosting the receive signal power, as well as waveform superposition property with over-the-air computation (AirComp) for fast model aggregation. By considering a practical scenario where high-dimensional local model updates are transmitted across multiple communication blocks, we characterize the convergence behaviors of the differentially private federated optimization algorithm. We further formulate a system optimization problem to optimize the learning accuracy while satisfying privacy and power constraints via the joint design of transmit power, artificial noise, and phase shifts at RIS, for which a two-step alternating minimization framework is developed. Simulation results validate our systematic, theoretical, and algorithmic achievements and demonstrate that RIS can achieve a better trade-off between privacy and accuracy for over-the-air FL systems.