With continuous expansion of satellite applications, the requirements for satellite communication services, such as communication delay, transmission bandwidth, transmission power consumption, and communication coverage, are becoming higher. This paper first presents an overview of the current development status of Low Earth Orbit (LEO) satellite constellations, and then conducts a demand analysis for multi-satellite data transmission based on LEO satellite constellations. The problem is described, and the challenges and difficulties of the problem are analyzed accordingly. On this basis, a multi-satellite data-transmission mathematical model is then constructed. Combining classical heuristic allocating strategies on the features of the proposed model, with the reinforcement learning algorithm Deep Q-Network (DQN), a two-stage optimization framework based on heuristic and DQN is proposed. Finally, by taking into account the spatial and temporal distribution characteristics of satellite and facility resources, a multi-satellite scheduling instance dataset is generated. Experimental results validate the rationality and correctness of the DQN algorithm in solving the collaborative scheduling problem of multi-satellite data transmission.