Misinformation is a prevalent societal issue due to its potential high risks. Out-Of-Context (OOC) misinformation, where authentic images are repurposed with false text, is one of the easiest and most effective ways to mislead audiences. Current methods focus on assessing image-text consistency but lack convincing explanations for their judgments, which are essential for debunking misinformation. While Multimodal Large Language Models (MLLMs) have rich knowledge and innate capability for visual reasoning and explanation generation, they still lack sophistication in understanding and discovering the subtle cross-modal differences. In this paper, we introduce SNIFFER, a novel multimodal large language model specifically engineered for OOC misinformation detection and explanation. SNIFFER employs two-stage instruction tuning on InstructBLIP. The first stage refines the model's concept alignment of generic objects with news-domain entities and the second stage leverages OOC-specific instruction data generated by language-only GPT-4 to fine-tune the model's discriminatory powers. Enhanced by external tools and retrieval, SNIFFER not only detects inconsistencies between text and image but also utilizes external knowledge for contextual verification. Our experiments show that SNIFFER surpasses the original MLLM by over 40\% and outperforms state-of-the-art methods in detection accuracy. SNIFFER also provides accurate and persuasive explanations as validated by quantitative and human evaluations.