Coded aperture snapshot spectral imaging (CASSI) captures a full frame spectral image as a single compressive image and is mandatory to reconstruct the underlying hyperspectral image (HSI) from the snapshot as the post-processing, which is challenge inverse problem due to its ill-posed nature. Existing methods for HSI reconstruction from a snapshot usually employs optimization for solving the formulated image degradation model regularized with the empirically designed priors, and still cannot achieve enough reconstruction accuracy for real HS image analysis systems. Motivated by the recent advances of deep learning for different inverse problems, deep learning based HSI reconstruction method has attracted a lot of attention, and can boost the reconstruction performance. This study proposes a novel deep convolutional neural network (DCNN) based framework for effectively learning the spatial structure and spectral attribute in the underlying HSI with the reciprocal spatial and spectral modules. Further, to adaptively leverage the useful learned feature for better HSI image reconstruction, we integrate residual attention modules into our DCNN via exploring both spatial and spectral attention maps. Experimental results on two benchmark HSI datasets show that our method outperforms state-of-the-art methods in both quantitative values and visual effect.