Graph Neural Networks (GNNs) resurge as a trending research subject owing to their impressive ability to capture representations from graph-structured data. However, the black-box nature of GNNs presents a significant challenge in terms of comprehending and trusting these models, thereby limiting their practical applications in mission-critical scenarios. Although there has been substantial progress in the field of explaining GNNs in recent years, the majority of these studies are centered on static graphs, leaving the explanation of dynamic GNNs largely unexplored. Dynamic GNNs, with their ever-evolving graph structures, pose a unique challenge and require additional efforts to effectively capture temporal dependencies and structural relationships. To address this challenge, we present DyExplainer, a novel approach to explaining dynamic GNNs on the fly. DyExplainer trains a dynamic GNN backbone to extract representations of the graph at each snapshot, while simultaneously exploring structural relationships and temporal dependencies through a sparse attention technique. To preserve the desired properties of the explanation, such as structural consistency and temporal continuity, we augment our approach with contrastive learning techniques to provide priori-guided regularization. To model longer-term temporal dependencies, we develop a buffer-based live-updating scheme for training. The results of our extensive experiments on various datasets demonstrate the superiority of DyExplainer, not only providing faithful explainability of the model predictions but also significantly improving the model prediction accuracy, as evidenced in the link prediction task.