The rapid spread of misinformation, driven by digital media and AI-generated content, has made automatic claim verification essential. Traditional methods, which depend on expert-annotated evidence, are labor-intensive and not scalable. Although recent automated systems have improved, they still struggle with complex claims that require nuanced reasoning. To address this, we propose CRAVE, a Conflicting Reasoning Approach for explainable claim VErification, that verify the complex claims based on the conflicting rationales reasoned by large language models (LLMs). Specifically, CRAVE introduces a three-module framework. Ambiguity Elimination enchanced Evidence Retrieval module performs ambiguity elimination and entity-based search to gather relevant evidence related to claim verification from external sources like Wikipedia. Conflicting Perspective Reasoning and Preliminary Judgment module with LLMs adopts LLMs to reason rationales with conflicting stances about claim verification from retrieved evidence across four dimensions, i.e., direct evidence, semantic relationships, linguistic patterns, and logical reasoning and make a preliminary judgment. Finally, Small Language Model (SLM) based Judge module is fine-tuned to make use of preliminary judgment from LLMs to assess the confidence of the conflicting rationales and make a final authenticity judgment. This methodology allows CRAVE to capture subtle inconsistencies in complex claims, improving both the accuracy and transparency of claim verification. Extensive experiments on two public claim verification datasets demonstrate that our CRAVE model achieves much better performance than state-of-the-art methods and exhibits a superior capacity for finding relevant evidence and explaining the model predictions. The code is provided at https://github.com/8zym/CRAVE.