Snapshot compressive spectral imaging (SCI) is a potential technology in the field of hyperspectral imaging (HSI), where multi-frame spectral images are compressed into a single snapshot measurement and collected by a 2D detector. Existing reconstruction algorithms of SCI systems do not make full use of the redundancy of hyperspectral data, artifacts and blur degrade the quality of the reconstruction target image. In this paper, an efficient algorithm, named as spatial structural sparsity and spectral low-rank priors (SSS-SLR), is proposed based on two inherent priors of hyperspectral images. Specifically, the spatial structural sparsity and spectral low-rank priori are simultaneously integrated into the SCI image restoration process to establish a variational optimization model, which can be solved via an alternating minimization algorithm. Extensive reconstruction results of hyperspectral data cube from both synthetic and real datasets demonstrate that the proposed method significantly outperforms the canonical algorithms.