Session-based recommendation (SR) has gained increasing attention in recent years. Quite a great amount of studies have been devoted to designing complex algorithms to improve recommendation performance, where deep learning methods account for the majority. However, most of these methods are black-box ones and ignore to provide moderate explanations to facilitate users' understanding, which thus might lead to lowered user satisfaction and reduced system revenues. Therefore, in our study, we propose a generic Reinforced Explainable framework with Knowledge graph for Session-based recommendation (i.e., REKS), which strives to improve the existing black-box SR models (denoted as non-explainable ones) with Markov decision process. In particular, we construct a knowledge graph with session behaviors and treat SR models as part of the policy network of Markov decision process. Based on our particularly designed state vector, reward strategy, and loss function, the reinforcement learning (RL)-based framework not only achieves improved recommendation accuracy, but also provides appropriate explanations at the same time. Finally, we instantiate the REKS in five representative, state-of-the-art SR models (i.e., GRU4REC, NARM, SR-GNN, GCSAN, BERT4REC), whereby extensive experiments towards these methods on four datasets demonstrate the effectiveness of our framework on both recommendation and explanation tasks.