Indonesian claim verification faces challenges in producing transparent and evidence-grounded explanations that mitigate unsupported or hallucinated reasoning by explicitly grounding model decisions in relevant evidence. This study proposes an evidence-based verification framework that integrates query generation, evidence selection, and LLM-driven explanation generation. Using the Indonesian X-FACT dataset, we conduct the first systematic comparison of explanation styles and evaluate their impact on reasoning quality and veracity prediction. Experimental results show that the proposed method achieves the best overall performance, reaching 0.92 accuracy and 0.87 F1-score, outperforming ungrounded explanation baselines. Explanation quality is evaluated through human assessment, yielding almost perfect inter-annotator agreement (Gwet's AC2 = 0.8093–0.8791) across four quality dimensions including faithfulness, logical flow, fluency, and informativeness. These findings demonstrate that explicit evidence yields more reliable verification outcomes and establishes a strong benchmark for explainable Indonesian claim verification. The implementation code, prompt templates, and dataset are available at the following repository (https://github.com/ridhaarrahmi/evidence-based-explanation.git). This article is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. License details: https://creativecommons.org/licenses/by-sa/4.0/