We propose a triple-frequency Beidou navigation satellite system (BDS) cycle-slip detection and repair method that balances accuracy and interpretability. It addresses the limited sensitivity of traditional approaches under diverse cycle-slip scenarios. We construct three detection observables from two geometry-free phase combinations (GFCs) and one pseudorange-phase combination (PPC), and augment them with sliding-window statistics as input features. We train a Kolmogorov-Arnold network (KAN) with Huber loss to learn the slip-free trend. We then apply robust residual-based thresholding to detect and repair slips. To align model behavior with physical mechanisms and enhance interpretability, we develop a SHAP-KAN framework that quantifies feature contributions and visualizes KAN's internal spline mappings. With the same number of hidden units, our method outperforms a backpropagation (BP) baseline: MAE/RMSE decrease by ~32%/~31%, R² increases by ~0.21, and runtime decreases by ~22%. Across satellites, KAN outputs exhibit marked noise reduction and enable tighter decision thresholds. All slip types are detected with no missed detections, and post-repair biases are < 0.1 cycles, consistent with the injected values. Overall, the method achieves accurate cycle-slip detection and repair for triple-frequency BDS and provides a physically consistent interpretation of the model outputs.