Space-air-ground integrated networks (SAGINs) help enhance the service performance in the sixth generation communication system. SAGIN is basically composed of satellites, aerial vehicles, ground facilities, as well as multiple terrestrial users. Therein, the low earth orbit (LEO) satellites are popular in recent years due to the low cost of development and launch, global coverage and delay-enabled services. Moreover, LEO satellites can support various applications, e.g., direct access, relay, caching and computation. In this work, we firstly provide the preliminaries and framework of SAGIN, in which the characteristics of LEO satellites, high altitude platforms, as well as unmanned aerial vehicles are analyzed. Then, the roles and potentials of LEO satellite in SAGIN are analyzed for access services. A couple of advanced techniques such as multi-access edge computing (MEC) and network function virtualization are introduced to enhance the LEO-based access service abilities as hierarchical MEC and network slicing in SAGIN. In addition, corresponding use cases are provided to verify the propositions. Besides, we also discuss the open issues and promising directions in LEO-enabled SAGIN access services for the future research.