Reliable and flexible emergency networks are of paramount essence for disaster relief during or after the event of disasters due to the destruction or lack of terrestrial communication infrastructures. Thanks to fast deployment and flexible mobilities, unmanned aerial vehicles (UAVs) emerge as a promising paradigm to efficiently establish emergency networks and perform immediate disaster relief tasks in affected areas. However, in such UAV-assisted disaster relief networks (UDRNs), the limited onboard batteries and computational capacities of UAVs hinder them from performing computation-intensive missions. Moreover, critical security vulnerabilities arise in data transmission among UAVs owing to the untrusted environment, open communication channels, and unreliable misbehavior tracing. To this end, this article investigates UDRNs based on blockchain and machine learning to achieve secure and efficient data transmission. Specifically, we first present a lightweight blockchain-enabled collaborative aerial-ground networking framework to safeguard data delivery under disasters, where a credit-based delegated proof-of-stake consensus protocol is further devised to enhance consensus efficiency while promoting UAVs' honest behaviors. In addition, by harnessing the idle computing power of ground vehicles (referred to as vehicular fog computing), a novel reinforcement learning-based algorithm is developed to intelligently offload UAVs' computation missions to the moving vehicles in the dynamic environment. Experimental results demonstrate that the proposed framework outperforms the existing approaches, in terms of consensus security, user utility, task latency, and energy consumption. Finally, future research directions in this emerging area are discussed.