Driven by the ever-increasing penetration and proliferation of data-driven applications, a new generation of wireless communication, the sixth-generation (6G) mobile system enhanced by artificial intelligence (AI), has attracted substantial research interests. Among various candidate technologies of 6G, low earth orbit (LEO) satellites have appealing characteristics of ubiquitous wireless access. However, the costs of satellite communication (SatCom) are still high, relative to counterparts of ground mobile networks. To support massively interconnected devices with intelligent adaptive learning and reduce expensive traffic in SatCom, we propose federated learning (FL) in LEO-based satellite communication networks. We first review the state-of-the-art LEO-based SatCom and related machine learning (ML) techniques, and then analyze four possible ways of combining ML with satellite networks. The learning performance of the proposed strategies is evaluated by simulation and results reveal that FL-based computing networks improve the performance of communication overheads and latency. Finally, we discuss future research topics along this research direction.