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Understanding sources of a model's uncertainty regarding its predictions is crucial for effective human-AI collaboration. Prior work proposes to use numerical uncertainty or hedges (I'm not sure, but...''), which do not explain uncertainty arising from conflicting evidence, leaving users unable to resolve disagreements or rely on the output. We introduce CLUE (Conflict-&Agreement-aware Language-model Uncertainty Explanations), the first framework to generate natural language explanations of model uncertainty by: (i) identifying relationships between spans of text that expose claim-evidence or inter-evidence conflicts/agreements driving the model's predictive uncertainty in an unsupervised way; and (ii) generating explanations via prompting and attention steering to verbalize these critical interactions. Across three language models and two fact-checking datasets, we demonstrate that CLUE generates explanations that are more faithful to model uncertainty and more consistent with fact-checking decisions than prompting for explanation of uncertainty without span-interaction guidance. Human evaluators find our explanations more helpful, more informative, less redundant, and better logically aligned with the input than this prompting baseline. CLUE requires no fine-tuning or architectural changes, making it plug-and-play for any white-box language model. By explicitly linking uncertainty to evidence conflicts, it offers practical support for fact-checking and readily generalizes to other tasks that require reasoning over complex information.