Unmanned-aerial-vehicle (UAV) enabled radio access is an effective technology to improve the wireless coverage, in particular for remote and disaster-struck areas. It will become a key enabler in the forthcoming 5G heterogeneous cellular networks to provide improved and resilient coverage. In this article, we study edge caching for multiple UAV-enabled radio access networks (UAV-RANs) and investigate how the overall spectral efficiency (SE) can be improved by efficient edge caching. In this context, UAV base stations (UBSs) may not serve the immediate close-by users, but serve the users based on the requested contents, which brings the service contents closer to users. Based on analyses of the SE achieved by the content-centric UAV-RAN, a hybrid caching strategy is proposed to further improve the SE. In order to cache more files with limited cache resources while guaranteeing the quality-of-service, the contents are divided into two subsets, a popular set and a less popular set, based on their popularity profile that follows Zipf distribution. The popular files, according to the proposed hybrid caching strategy, are cached at all the UBSs, while each less popular file is cached at one UBS only. The overall SE is maximized by finding an optimal popularity threshold between the two subsets. By relaxing the formulated problem, analytical expression for the optimized threshold is derived. The effectiveness of the proposed method is verified by numerical examples where existing benchmark schemes are compared and outperformed.