In this paper, we consider a cooperative communication network where multiple low-Earth-orbit (LEO) satellites provide services to multiple ground users (GUs) cooperatively at the same time and on the same frequency. The multi-satellite cooperation has great potential in extending communication coverage and increasing spectral efficiency. Considering that the on-board radio-frequency circuit resources and computation resources on each satellite are restricted, we aim to propose a low-complexity yet efficient multi-satellite cooperative transmission framework. Specifically, we first propose a hybrid beamforming method consisting of analog beamforming for beam alignment and digital beamforming for interference mitigation. Then, to establish appropriate connections between the satellites and GUs, we propose a heuristic user scheduling algorithm which determines the connections according to the total spectral efficiency increment of the multi-satellite cooperative network. Next, considering the intrinsic connection between beamforming and user scheduling, a joint hybrid beamforming and user scheduling (JHU) scheme is proposed to dramatically improve the performance of the multi-satellite cooperative network. In addition to the single-connection scenario, we also consider the multi-connection case using the JHU scheme. Extensive simulations conducted over different LEO satellite constellations and across various GU locations demonstrate the superiority of the proposed schemes in both overall and per-user spectral efficiencies.