Multi-connectivity (MC) in satellite-terrestrial integrated networks (STINs), included in 3GPP standards, is regarded as a promising technology for future networks. The significant advantages of MC in improving coverage, communication, and sensing through satellite-terrestrial collaboration have sparked widespread interest. In this article, we first introduce three fundamental deployment architectures of MC systems in STINs, including multi-satellite, single-satellite single-base-station, and multi-satellite multi-base-station configurations. Considering the emerging but still evolving satellite networking, we explore system design challenges such as satellite networking schemes, e.g., cell-free and multi-tier satellite networks. Then, key technical challenges that severely influence the quality of mutual communications, including beamforming, channel estimation, and synchronization, are discussed subsequently. Furthermore, typical applications such as coverage enhancement, traffic offloading, collaborative sensing, and low-altitude communication are demonstrated, followed by a case study comparing coverage performance in MC and single-connectivity (SC) configurations. Several essential future research directions for MC in STINs are presented to facilitate further exploration.