Artificial Intelligence (AI) is rapidly embedded in critical decision-making systems, however their foundational ``black-box'' models require eXplainable AI (XAI) solutions to enhance transparency, which are mostly oriented to experts, making no sense to non-experts. Alarming evidence about AI's unprecedented human values risks brings forward the imperative need for transparent human-centered XAI solutions. In this work, we introduce a domain-, model-, explanation-agnostic, generalizable and reproducible framework that ensures both transparency and human-centered explanations tailored to the needs of both experts and non-experts. The framework leverages Large Language Models (LLMs) and employs in-context learning to convey domain- and explainability-relevant contextual knowledge into LLMs. Through its structured prompt and system setting, our framework encapsulates in one response explanations understandable by non-experts and technical information to experts, all grounded in domain and explainability principles. To demonstrate the effectiveness of our framework, we establish a ground-truth contextual ``thesaurus'' through a rigorous benchmarking with over 40 data, model, and XAI combinations for an explainable clustering analysis of a well-being scenario. Through a comprehensive quality and human-friendliness evaluation of our framework's explanations, we prove high content quality through strong correlations with ground-truth explanations (Spearman rank correlation=0.92) and improved interpretability and human-friendliness to non-experts through a user study (N=56). Our overall evaluation confirms trust in LLMs as HCXAI enablers, as our framework bridges the above Gaps by delivering (i) high-quality technical explanations aligned with foundational XAI methods and (ii) clear, efficient, and interpretable human-centered explanations for non-experts.