To address the time-consuming and complex multiple agile Earth observation satellite scheduling problem (multiAEOSSP), we propose LLM-EC, an LLM-assisted evolutionary task dispatching heuristic design framework with three key innovations: 1) Customized prompt strategies for initialization, crossover, and mutation tailored for efficient order-dispatching heuristics. 2) The evolved heuristics decompose multi-AEOSSP into parallel single-satellite subproblems, significantly reducing solution complexity. 3) Experiments on realistic scenarios demonstrate that LLM-EC outperforms both standard LLMbased evolutionary algorithms and expert-designed heuristics in solution quality and convergence speed.