The prevalence of fake news on social media calls for automated fact-checking systems that deliver not only accurate verdicts but also faithful explanations. However, existing large language model (LLM)-based methods often overlook deceptive misinformation styles in generated explanations, producing unfaithful rationales that may mislead human judgment. They also rely heavily on external knowledge sources, which can introduce hallucinations and incur substantial latency, undermining both reliability and responsiveness in real-time settings. To address these limitations, we propose REason-guided Fact-checking with Latent EXplanations (REFLEX), a self-refining framework that explicitly controls reasoning style by anchoring explanations to the predicted verdict. REFLEX leverages self-disagreement veracity signals between a backbone model and its fine-tuned variant to construct steering vectors, thereby naturally disentangling factual content from stylistic cues. Experiments on a real-world benchmark show that REFLEX achieves state-of-the-art performance under LLaMA-series models using only 465 self-refined samples. Owing to its transferability, REFLEX also yields gains of up to 7.54 Macro-F1 points on in-the-wild data. Further analysis shows that our method effectively mitigates faithful hallucination, leading to both more reliable explanations and more accurate verdicts than prior explainable fact-checking approaches.