Reliable prediction of post-initiation run-out distance of rainfall-induced landslides is essential for hazard assessment, evacuation planning, and disaster-risk mitigation. However, most existing data-driven approaches formulate run-out prediction as a deterministic regression problem and therefore provide limited information on predictive uncertainty, rare long-runout events, and explainable decision support. To address these limitations, this study proposes CQR-EVT-XAI, a trustworthy AI framework that integrates Quantile LightGBM, Conformalized Quantile Regression (CQR), Extreme Value Theory (EVT), and Explainable Artificial Intelligence (XAI) for uncertainty-aware and explainable landslide run-out risk prediction. Based on 10,158 rainfall-induced landslide samples, physics-informed features are constructed from elevation difference H, source area A, source volume V, and mean slope angle $\theta$. The proposed framework generates calibrated prediction intervals, threshold-based exceedance probabilities, upper-tail risk indicators, and interpretable risk levels. The CQR-LightGBM median model achieves high point-prediction accuracy, with $R^2$ = 0.939, RMSE = 18.03 m, and MAE = 6.55 m. Conformal calibration improves the empirical coverage of the nominal 90% and 95% prediction intervals from 0.813 to 0.903 and from 0.876 to 0.953, respectively. Tail-risk analysis shows that the upper prediction bound ${\widehat{L}}_{95}$ effectively identifies extreme long-runout events, achieving recall values of 0.974 and 0.900 for L > 300 m and L > 500 m, respectively. SHAP analysis reveals that elevation difference H, source volume V, and energy-related derived features dominate both median run-out prediction and upper-tail risk behavior, while slope-related variables mainly influence predictive uncertainty and exceedance-risk levels. These results demonstrate that the proposed CQR-EVT-XAI framework provides a practical workflow for calibrated uncertainty quantification, tail-risk identification, and explainable decision support in rainfall-induced landslide run-out risk assessment.