With the construction of low-earth orbit (LEO) satellite constellations, ubiquitous connectivity has been achieved. Terrestrial networks (TNs), such as cellular networks, are mainly deployed in specific urban areas and use licensed spectrum. However, in remote areas where terrestrial infrastructure is sparse, licensed spectrum bands are often underutilized. To accommodate the increasing communication needs, non-terrestrial networks (NTNs) can opportunistically access this idle spectrum to improve spectrum efficiency via spectrum sharing (SS). Therefore, bringing NTNs to a shared spectrum with TNs can improve network capacity under reasonable interference management. In satellite-terrestrial integrated networks (STINs), the comprehensive coverage of a satellite and the unbalanced communication resources of STINs make it challenging to manage mutual interference between NTN and TN effectively. This article presents the fundamentals and prospects of SS in STINs by introducing four SS frameworks, their potential application scenarios, and technical challenges. Furthermore, advanced SS approaches related to interference management in STINs and performance metrics of SS in STINs are introduced. Moreover, a preliminary performance evaluation showcases the potential for sharing the spectrum between NTN and TN. Finally, future research opportunities for SS in STINs are discussed.