The space-air-ground integrated network (SAGIN), one of the key technologies for next-generation mobile communication systems, can facilitate data transmission for users all over the world, especially in some remote areas where vast amounts of informative data are collected by Internet of remote things (IoRT) devices to support various data-driven artificial intelligence (AI) services. However, training AI models centrally with the assistance of SAGIN faces the challenges of highly constrained network topology, inefficient data transmission, and privacy issues. To tackle these challenges, we first propose a novel topology-aware federated learning framework for the SAGIN, namely Olive Branch Learning (OBL). Specifically, the IoRT devices in the ground layer leverage their private data to perform model training locally, while the air nodes in the air layer and the ring-structured low earth orbit (LEO) satellite constellation in the space layer are in charge of model aggregation (synchronization) at different scales.To further enhance communication efficiency and inference performance of OBL, an efficient Communication and Non-IID-aware Air node-Satellite Assignment (CNASA) algorithm is designed by taking the data class distribution of the air nodes as well as their geographic locations into account. Furthermore, we extend our OBL framework and CNASA algorithm to adapt to more complex multi-orbit satellite networks. We analyze the convergence of our OBL framework and conclude that the CNASA algorithm contributes to the fast convergence of the global model. Extensive experiments based on realistic datasets corroborate the superior performance of our algorithm over the benchmark policies.