The unmanned aerial vehicle (UAV) system, with its advantages of flexible deployment and line-of-sight propagation, has become an essential tool for assisting mobile communications in handling high-density data processing and emergency communications. However, the computational processing capabilities and endurance issues of UAVs under complex environments remain significant technological bottlenecks. The development of mobile edge computing (MEC) technology provides an effective solution to address UAVs' computational and energy consumption challenges. A distributed task offloading strategy based on a multi-agent reinforcement learning algorithm was proposed for MEC-assisted UAV systems. The task offloading and resource allocation process of UAVs was modelled as a Markov game process (MGP) involving multiple MEC nodes. To solve the MGP problem, a distributed reinforcement learning algorithm for multi-agent collaboration was proposed. The algorithm enabled agents to find the optimal strategies through online collaborative learning based on local observation information. In comparative experiments, the convergence and system performance of the proposed scheme were evaluated. The results show that the proposed scheme outperforms the comparison schemes in terms of convergence speed, energy consumption, and unloading rate.