Recent advances in large language models and vision-language models have led to growing interest in explainable evaluation metrics for image captioning. However, these metrics generate explanations without standardized criteria, and the overall quality of the generated explanations remains unverified. In this paper, we propose EXPERT, a reference-free evaluation metric that provides structured explanations based on three fundamental criteria: fluency, relevance, and descriptiveness. By constructing large-scale datasets of high-quality structured explanations, we develop a two-stage evaluation template to effectively supervise a vision-language model for both scoring and explanation generation. EXPERT achieves state-of-the-art results on benchmark datasets while providing significantly higher-quality explanations than existing metrics, as validated through comprehensive human evaluation. Our code and datasets are available at https://github.com/hjkim811/EXPERT.