The advancement of nanosatellite techniques has boosted the growth of satellite-originated data and applications. Satellite edge computing (SEC) is envisioned to provide in-orbit processing of the sensed data to save the scarce terrestrial-satellite communication resources and support mission-critical services. While most of the existing SEC studies mainly focus on general computing tasks, we present a two-tier collaborative processing framework for the important and unique hyperspectral image (HSI) processing task. Our framework carefully selects bands out of the collected HSIs and sends them back for further analysis. We first conduct a comprehensive data analysis to reveal the non-trivial relationship between the band selection and the eventual analytic performance. We then formulate the band selection problem in this collaborative setting as a utility maximization problem that jointly considers the analytic, energy, and communication factors. A novel multi-agent reinforcement learning approach, named MaHSI, is proposed to solve it in the dynamic SEC environment. Our multi-agent design judiciously embeds the complex correlations among bands as collaborations among agents and significantly reduces the exploration space. Extensive experiments on real-world HSI datasets prove that our approach not only outperforms the existing classical band selection algorithms in accuracy and inference speed but also brings the highest utility to the satellites.