Recent commonsense-reasoning tasks are typically discriminative in nature, where a model answers a multiple-choice question for a certain context. Discriminative tasks are limiting because they fail to adequately evaluate the model's ability to reason and explain predictions with underlying commonsense knowledge. They also allow such models to use reasoning shortcuts and not be "right for the right reasons". In this work, we present ExplaGraphs, a new generative and structured commonsense-reasoning task (and an associated dataset) of explanation graph generation for stance prediction. Specifically, given a belief and an argument, a model has to predict whether the argument supports or counters the belief and also generate a commonsense-augmented graph that serves as non-trivial, complete, and unambiguous explanation for the predicted stance. The explanation graphs for our dataset are collected via crowdsourcing through a novel Collect-Judge-And-Refine graph collection framework that improves the graph quality via multiple rounds of verification and refinement. A significant 83% of our graphs contain external commonsense nodes with diverse structures and reasoning depths. We also propose a multi-level evaluation framework that checks for the structural and semantic correctness of the generated graphs and their plausibility with human-written graphs. We experiment with state-of-the-art text generation models like BART and T5 to generate explanation graphs and observe that there is a large gap with human performance, thereby encouraging useful future work for this new commonsense graph-based explanation generation task.