Commonsense explanation generation aims to empower the machine's sense-making capability by generating plausible explanations to statements against commonsense. While this task is easy to human, the machine still struggles to generate reasonable and informative explanations. In this work, we propose a method that first extracts the underlying concepts which are served as bridges in the reasoning chain and then integrates these concepts to generate the final explanation. To facilitate the reasoning process, we utilize external commonsense knowledge to build the connection between a statement and the bridge concepts by extracting and pruning multi-hop paths to build a subgraph. We design a bridge concept extraction model that first scores the triples, routes the paths in the subgraph, and further selects bridge concepts with weak supervision at both the triple level and the concept level. We conduct experiments on the commonsense explanation generation task and our model outperforms the state-of-the-art baselines in both automatic and human evaluation.