Multimodal Large Language Models struggle to maintain reliable performance under extreme real-world visual degradations, which impede their practical robustness. Current robustness enhancement methods rely on implicit training/adaptation that focuses solely on visual encoder generalization, suffering from limited interpretability and isolated optimization. To overcome these limitations, we propose Robust-R1, a novel framework that explicitly models visual degradations through structured reasoning chains. Our approach integrates: (i) supervised fine-tuning for degradation-aware reasoning foundations, (ii) reward-driven alignment for accurately perceiving degradation parameters, and (iii) dynamic reasoning depth scaling adapted to degradation intensity. To support this methodology, we introduce a novel 11K dataset featuring realistic degradations synthesized across four critical real-world visual processing stages, each annotated with structured chains connecting degradation parameters, perceptual effects, and pristine semantic reasoning. Comprehensive evaluations demonstrate state-of-the-art robustness: Robust-R1 outperforms all general and robust baselines on the real-world degradation benchmark R-Bench, while maintaining superior anti-degradation performance under multi-intensity adversarial degradations on MMBench, MMStar, and RealWorldQA. We will release our code, demo, and dataset soon.