Satellite-terrestrial networks (STNs) are anticipated to deliver seamless IoT services across expansive regions. Given the constrained resources available for offloading computationally intensive tasks like video streaming, it is crucial to establish collaborative computing among diverse components within STNs. In this paper, we present the task offloading challenge as a multi-objective optimization problem, leveraging the collaboration between ground devices/users and satellites. We propose a collaborative computing scheme that optimally assigns computing tasks to various nodes within STNs to enhance service performance including quality of experience (QoE). This algorithm initially dynamically selects an end-to-end path that balances service time and resource utilization. For each selected path, a multi-agent soft actor-critic (MA-SAC)-based algorithm is introduced to make adaptive decisions and collaboratively assign optimal heterogeneous resources to the given computing tasks. In this algorithm, the ground station bridging satellite network and terrestrial network is treated as agent to extract the information from both STNs and users. Through MA-SAC, multiple agents cooperate to determine the adaptive bitrate and network resources for the arriving tasks. The numerical results demonstrate that our proposal outperforms comparative schemes across various computing tasks in terms of various criteria.