Fact-checking research has extensively explored verification but less so the generation of natural-language explanations, crucial for user trust. While Large Language Models (LLMs) excel in text generation, their capability for producing faithful explanations in fact-checking remains underexamined. Our study investigates LLMs' ability to generate such explanations, finding that zero-shot prompts often result in unfaithfulness. To address these challenges, we propose the Multi-Agent Debate Refinement (MADR) framework, leveraging multiple LLMs as agents with diverse roles in an iterative refining process aimed at enhancing faithfulness in generated explanations. MADR ensures that the final explanation undergoes rigorous validation, significantly reducing the likelihood of unfaithful elements and aligning closely with the provided evidence. Experimental results demonstrate that MADR significantly improves the faithfulness of LLM-generated explanations to the evidence, advancing the credibility and trustworthiness of these explanations.