Recent advances in eXplainable AI (XAI) for education have highlighted a critical challenge: ensuring that explanations for state-of-the-art AI models are understandable for non-technical users such as educators and students. In response, we introduce iLLuMinaTE, a zero-shot, chain-of-prompts LLM-XAI pipeline inspired by Miller's cognitive model of explanation. iLLuMinaTE is designed to deliver theory-driven, actionable feedback to students in online courses. iLLuMinaTE navigates three main stages - causal connection, explanation selection, and explanation presentation - with variations drawing from eight social science theories (e.g. Abnormal Conditions, Pearl's Model of Explanation, Necessity and Robustness Selection, Contrastive Explanation). We extensively evaluate 21,915 natural language explanations of iLLuMinaTE extracted from three LLMs (GPT-4o, Gemma2-9B, Llama3-70B), with three different underlying XAI methods (LIME, Counterfactuals, MC-LIME), across students from three diverse online courses. Our evaluation involves analyses of explanation alignment to the social science theory, understandability of the explanation, and a real-world user preference study with 114 university students containing a novel actionability simulation. We find that students prefer iLLuMinaTE explanations over traditional explainers 89.52% of the time. Our work provides a robust, ready-to-use framework for effectively communicating hybrid XAI-driven insights in education, with significant generalization potential for other human-centric fields.