Computer-aided early diagnosis of Alzheimer's disease (AD) and its prodromal form mild cognitive impairment (MCI) based on structure Magnetic Resonance Imaging (sMRI) has provided a cost-effective and objective way for early prevention and treatment of disease progression, leading to improved patient care. In this work, we have proposed a novel computer-aided approach for early diagnosis of AD by introducing an explainable 3D Residual Attention Deep Neural Network (3D ResAttNet) for end-to-end learning from sMRI scans. Different from the existing approaches, the novelty of our approach is three-fold: 1) A Residual Self-Attention Deep Neural Network has been proposed to capture local, global and spatial information of MR images to improve diagnostic performance; 2) An explainable method using Gradient-based Localization Class Activation mapping (Grad-CAM) has been introduced to improve the interpretability of the proposed method; 3) This work has provided a full end-to-end learning solution for automated disease diagnosis. Our proposed 3D ResAttNet method has been evaluated on a large cohort of subjects from real datasets for two changeling classification tasks (i.e. Alzheimer's disease (AD) vs. Normal cohort (NC) and progressive MCI (pMCI) vs. stable MCI (sMCI)). The experimental results show that the proposed approach has a competitive advantage over the state-of-the-art models in terms of accuracy performance and generalizability. The explainable mechanism in our approach is able to identify and highlight the contribution of the important brain parts (e.g., hippocampus, lateral ventricle and most parts of the cortex) for transparent decisions.