The past few years have witnessed a tremendous increase on the use of unmanned aerial vehicles (UAVs) in civilian applications, which call for high-performance communication between UAVs and their ground clients, especially when they are densely deployed. To achieve this goal, cellular base stations (BSs) can be leveraged to provide a new and promising solution to support massive UAV communications simultaneously in a cost-effective way. However, different from terrestrial communication channels, UAV-to-BS channels are usually dominated by the light-of-sight (LoS) link, which aggravates the co-channel interference and renders the spatial frequency reuse in existing cellular networks ineffective. In this paper, we consider the use of a directional antenna at each UAV to confine the interference to/from other UAV users within a limited region and hence improve the spatial reuse of the spectrum. Under this model, a UAV user may be temporarily blocked from communication if it cannot find any BS in its antenna main-lobe, or it finds that all BSs under its main-lobe are simultaneously covered by those of some other UAVs and hence suffer from strong co-channel interference. Assuming independent homogeneous Poisson point processes (HPPPs) for the UAVs' and ground BSs' locations respectively, we first analytically derive a closed-form upper bound for the UAV blocking probability and then characterize the achievable average spatial throughput of the cellular-enabled UAV communication network, in terms of various key parameters including the BS/UAV densities as well as the UAV's flying altitude and antenna beamwidth. Simulation results verify that the derived bound is practically tight, and further show that adaptively adjusting the UAV altitude and/or beamwidth with different BS/UAV densities can significantly reduce the UAV blocking probability and hence improve the network spatial throughput.