The rapid surge of multimedia and video applications poses challenges to the content-delivering service in wireless networks. In this paper, we study the proactive caching problem in unmanned aerial vehicle (UAV)-enabled networks, where a number of UAVs are deployed to offer content delivery service for user equipments (UEs). In order to acquire user request information, we first propose a bidirectional long short-term memory-based user request prediction algorithm. Then, based on the obtained user content requests, we examine the content fetching delay of users and formulate UAV deployment, content caching, and power allocation problems as an overall content fetching delay minimization problem. To solve the formulated optimization problem, we decouple it into two subproblems, namely, the UAV deployment and content caching subproblem, and the power allocation subproblem, and solve the two subproblems by using an alternate iteration-based algorithm. Specifically, we first design a modified K-means-based clustering scheme to group UEs into various clusters, and then develop a UAV deployment strategy for individual clusters by applying quadratic transformation and first-order Taylor expansion. A heuristic proactive content caching algorithm is further proposed for individual UAVs. Finally, the Lagrangian dual method is employed to solve the power allocation subproblem. Simulation results demonstrate the effectiveness of the proposed algorithms.