Machine learning (ML) is becoming increasingly popular in meteorological decision-making. Although the literature on explainable artificial intelligence (XAI) is growing steadily, user-centered XAI studies have not extend to this domain yet. This study defines three requirements for explanations of black-box models in meteorology through user studies: statistical model performance for different rainfall scenarios to identify model bias, model reasoning, and the confidence of model outputs. Appropriate XAI methods are mapped to each requirement, and the generated explanations are tested quantitatively and qualitatively. An XAI interface system is designed based on user feedback. The results indicate that the explanations increase decision utility and user trust. Users prefer intuitive explanations over those based on XAI algorithms even for potentially easy-to-recognize examples. These findings can provide evidence for future research on user-centered XAI algorithms, as well as a basis to improve the usability of AI systems in practice.