Recommendation systems play a crucial role in various domains, suggesting items based on user behavior. And the lack of transparency in presenting recommendations can lead to user confusion. Thus, recommendation explanation methods are proposed to generate natural language explanations for users, which usually require intermediary representations of the recommendation model or need to conduct latent alignment training to the recommendation model. However, this additional training step usually causes potential performance issues due to the different training objectives between the recommendation task and the explanation task. In this paper, we introduce Data-level Recommendation Explanation (DRE), a non-intrusive explanation framework for black-box recommendation models. We propose a data-level alignment method, leveraging large language models to reason relationships between user data and recommended items, without any additional training or intermediary representations for the recommendation model. Additionally, we also address the challenge of enriching the details of the explanation by introducing target-aware user preference distillation, utilizing item reviews. Experimental results on several benchmark datasets demonstrate the effectiveness of the DRE in providing accurate and user-centric explanations, enhancing user engagement with recommended items.