Empathetic dialogue requires not only recognizing a user's emotional state but also making strategy-aware, context-sensitive decisions throughout response generation. However, the lack of a comprehensive empathy strategy framework, explicit task-aligned multi-stage reasoning, and high-quality strategy-aware data fundamentally limits existing approaches, preventing them from effectively modeling empathetic dialogue as a complex, multi-stage cognitive and decision-making process. To address these challenges, we propose STRIDE-ED, a STRategy-grounded, Interpretable, and DEep reasoning framework that models Empathetic Dialogue through structured, strategy-conditioned reasoning. To support effective learning, we develop a strategy-aware data refinement pipeline integrating LLM-based annotation, multi-model consistency-weighted evaluation, and dynamic sampling to construct high-quality training data aligned with empathetic strategies. Furthermore, we adopt a two-stage training paradigm that combines supervised fine-tuning with multi-objective reinforcement learning to better align model behaviors with target emotions, empathetic strategies, and response formats. Extensive experiments demonstrate that STRIDE-ED generalizes across diverse open-source LLMs and consistently outperforms existing methods on both automatic metrics and human evaluations. Our data and code are publicly available at https://github.com/jicoder-nwpu/STRIDE-ED.