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Effectiveness and interpretability are two essential properties for trustworthy AI systems. Most recent studies in visual reasoning are dedicated to improving the accuracy of predicted answers, and less attention is paid to explaining the rationales behind the decisions. As a result, they commonly take advantage of spurious biases instead of actually reasoning on the visual-textual data, and have yet developed the capability to explain their decision making by considering key information from both modalities. This paper aims to close the gap from three distinct perspectives: first, we define a new type of multi-modal explanations that explain the decisions by progressively traversing the reasoning process and grounding keywords in the images. We develop a functional program to sequentially execute different reasoning steps and construct a new dataset with 1,040,830 multi-modal explanations. Second, we identify the critical need to tightly couple important components across the visual and textual modalities for explaining the decisions, and propose a novel explanation generation method that explicitly models the pairwise correspondence between words and regions of interest. It improves the visual grounding capability by a considerable margin, resulting in enhanced interpretability and reasoning performance. Finally, with our new data and method, we perform extensive analyses to study the effectiveness of our explanation under different settings, including multi-task learning and transfer learning. Our code and data are available at https://github.com/szzexpoi/rex.