With the rapid development of large language models (LLMs), their application in the explainability of recommender systems has become a research hotspot. This study systematically reviews the research progress of LLMs in the explainability of recommender systems, providing a comprehensive overview covering current research status, evaluation metrics, datasets, and application scenarios. From a technical perspective, existing research is categorized into LLM-based recommender systems and LLM-aid recommender systems, further subdivided according to whether fine-tuning is required. In terms of evaluation metrics, manual evaluation and automated evaluation metrics are summarized, with automated evaluation metrics including traditional metrics, LLM-integrated metrics, and extended metrics. Moreover, the usage of public and private datasets is reviewed, with emphasis on the importance of review data in explainable recommendations. Finally, the practical applications of LLMs in the explainability of recommender systems across various domains are explored, and the challenges faced by current research as well as potential future research directions are analyzed.