Metaphor is not only a linguistic phenomenon but also reflects the concept projection between source and target domains in human cognition. Previous sequence tagging-based metaphor identification methods could not model the concept projection, resulting in a limitation that the outputs of these models are unexplainable in the predictions of the metaphoricity labels. In this work, we propose the first explainable metaphor identification model, inspired by Conceptual Metaphor Theory. The model is based on statistic learning, a lexical resource, and a novel reward mechanism. Our model can identify the metaphoricity on the word-pair level, and explain the predicted metaphoricity labels via learned concept mappings. The use of the reward mechanism allows the model to learn the optimal concept mappings without knowing their true labels. Our method is also applicable for the concepts that are out of training domains by using the lexical resource. The automatically generated concept mappings demonstrate the implicit human thoughts in metaphoric expressions. Our experiments show the effectiveness of the proposed model in metaphor identification, and concept mapping tasks, respectively.