Driven by the growing demand for intelligent remote sensing applications, large artificial intelligence (AI) models pre-trained on large-scale unlabeled datasets and fine-tuned for downstream tasks have significantly improved learning performance for various downstream tasks due to their generalization capabilities. However, many specific downstream tasks, such as extreme weather nowcasting (e.g., downburst and tornado), disaster monitoring, and battlefield surveillance, require real-time data processing. Traditional methods via transferring raw data to ground stations for processing often cause significant issues in terms of latency and trustworthiness. To address these challenges, satellite edge AI provides a paradigm shift from ground-based to on-board data processing by leveraging the integrated communication-and-computation capabilities in space computing power networks (Space-CPN), thereby enhancing the timeliness, effectiveness, and trustworthiness for remote sensing downstream tasks. Moreover, satellite edge large AI model (LAM) involves both the training (i.e., fine-tuning) and inference phases, where a key challenge lies in developing computation task decomposition principles to support scalable LAM deployment in resource-constrained space networks with time-varying topologies. In this article, we first propose a satellite federated fine-tuning architecture to split and deploy the modules of LAM over space and ground networks for efficient LAM fine-tuning. We then introduce a microservice-empowered satellite edge LAM inference architecture that virtualizes LAM components into lightweight microservices tailored for multi-task multimodal inference. Finally, we discuss the future directions for enhancing the efficiency and scalability of satellite edge LAM, including task-oriented communication, brain-inspired computing, and satellite edge AI network optimization.