Recently, a large number of Low Earth Orbit (LEO) satellites have been launched and deployed successfully in space by commercial companies, such as SpaceX. Due to multimodal sensors equipped by the LEO satellites, they serve not only for communication but also for various machine learning applications, such as space modulation recognition, remote sensing image classification, etc. However, the ground station (GS) may be incapable of downloading such a large volume of raw sensing data for centralized model training due to the limited contact time with LEO satellites (e.g. 5 minutes). Therefore, federated learning (FL) has emerged as the promising solution to address this problem via on-device training. Unfortunately, to enable FL on LEO satellites, we still face three critical challenges that are i) heterogeneous computing and memory capabilities, ii) limited uplink rate, and iii) model staleness. To this end, we propose FedSN as a general FL framework to tackle the above challenges, and fully explore data diversity on LEO satellites. Specifically, we first present a novel sub-structure scheme to enable heterogeneous local model training considering different computing, memory, and communication constraints on LEO satellites. Additionally, we propose a pseudo-synchronous model aggregation strategy to dynamically schedule model aggregation for compensating model staleness. To further demonstrate the effectiveness of the FedSN, we evaluate it using space modulation recognition and remote sensing image classification tasks by leveraging the data from real-world satellite networks. Extensive experimental results demonstrate that FedSN framework achieves higher accuracy, lower computing, and communication overhead than the state-of-the-art benchmarks and the effectiveness of each components in FedSN.