Devices located in remote regions often lack coverage from well-developed terrestrial communication infrastructure. This not only prevents them from experiencing high quality communication services but also hinders the delivery of machine learning services in remote regions. In this paper, we propose a new federated learning (FL) methodology tailored to space-air-ground integrated networks (SAGINs) to tackle this issue. Our approach strategically leverages the nodes within space and air layers as both (i) edge computing units and (ii) model aggregators during the FL process, addressing the challenges that arise from the limited computation powers of ground devices and the absence of terrestrial base stations in the target region. The key idea behind our methodology is the adaptive data offloading and handover procedures that incorporate various network dynamics in SAGINs, including the mobility, heterogeneous computation powers, and inconsistent coverage times of incoming satellites. We analyze the latency of our scheme and develop an adaptive data offloading optimizer, and also characterize the theoretical convergence bound of our proposed algorithm. Experimental results confirm the advantage of our SAGIN-assisted FL methodology in terms of training time and test accuracy compared with various baselines.