The rapid development of communication technologies in the past decades has provided immense vertical opportunities for individuals and enterprises. However, conventional terrestrial cellular networks have unfortunately neglected the huge geographical digital divide, since high-bandwidth wireless coverage is concentrated in urban areas. To meet the goal of 'connecting the unconnected,' integrating low Earth orbit (LEO) satellites with the terrestrial cellular networks has been widely considered as a promising solution. In this article, we first introduce the development roadmap of LEO satellite constellations (SatCons), including early attempts in LEO satellites with the emerging LEO constellations. Further, we discuss the unique opportunities of employing LEO SatCons for the delivery of integrating 5G networks. Specifically, we present their key performance indicators, which offer important guidelines for the design of associated enabling techniques, and then discuss the potential impact of integrating LEO SatCons with typical 5G use cases, where we engrave our vision of various vertical domains reshaped by LEO SatCons. Technical challenges are finally provided to specify future research directions.