In this paper, we investigate for the first time the benefits of wireless caching for the physical layer security (PLS) of wireless networks. In particular, a caching scheme enabling power-efficient PLS is proposed for cellular video streaming with constrained backhaul capacity. By sharing video data across a subset of base stations (BSs) through both caching and backhaul loading, secure cooperative transmission of several BSs is dynamically enabled in accordance with the cache status, the channel conditions, and the backhaul capacity. Thereby, caching reduces the data sharing overhead over the capacity-constrained backhaul links. More importantly, caching introduces additional secure degrees of freedom and enables a power-efficient design. We investigate the optimal caching and transmission policies for minimizing the total transmit power while providing quality of service (QoS) and guaranteeing secrecy during video delivery. A two-stage non-convex mixed-integer optimization problem is formulated, which optimizes the caching policy in an offline video caching stage and the cooperative transmission policy in an online video delivery stage. As the problem is NP-hard, suboptimal polynomial-time algorithms are proposed for low-complexity cache training and delivery control, respectively. Sufficient optimality conditions, under which the proposed schemes attain global optimal solutions, are also provided. Simulation results show that the proposed schemes achieve low secrecy outage probability and high power efficiency simultaneously.