Driven by the growing demand for intelligent remote sensing (RS) applications, large artificial intelligence (AI) models (LAMs) pretrained on large-scale unlabeled datasets and fine-tuned for downstream tasks have considerably improved learning performance for various downstream tasks because of their generalization capabilities. However, many specific downstream tasks, such as extreme weather nowcasting (e.g., downbursts and tornadoes), disaster monitoring, and battlefield surveillance, require real-time data processing. Traditional methods that transfer raw data to ground stations for processing often cause substantial 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, thereby enhancing the timeliness, effectiveness, and trustworthiness for RS downstream tasks. Moreover, satellite edge LAMs involve 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. Then, we introduce a microservice-empowered satellite edge LAM inference architecture that virtualizes LAM components into lightweight microservices tailored for multitask 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.