Satellite-terrestrial integrated networks have been proposed as a promising solution to provide global seamless coverage in next-generation communication networks. To enable the integration of satellite and terrestrial networks, network simulation is a fundamental and economically efficient step compared to the high cost of field trial and real-world system deployment. Simulation challenges arise with the fast growth of LEO mega-constellations, including frequent re-connection and handover, long satellite transmission delay, high dynamics of satellite network topologies, and the integration of heterogeneous infrastructures. More requirements emerge for satellite-terrestrial integrated simulation, including fidelity, scalability, extensibility, agility, and real-time. However, there is a lack of a state-of-the-art review for relevant simulators. To the best of our knowledge, this paper is the most comprehensive and latest survey that covers network simulators for satellite-terrestrial integrated networks, with all or partial simulation functionalities for satellite orbit simulation, physical layer modeling, and network protocols and algorithms. Compared with existing surveys, this survey contributes from three aspects: (1) an up-to-date collection and a comprehensive taxonomy of simulation tools in the past decade; (2) a summary of the main requirements and challenges; and (3) an inspiring summary for new research opportunities and publicly available simulation tools for follow-up research.