Deep neural network classifiers for dysarthria impairment severity face limitations regarding interpretability and treatment guidance. To overcome these, we introduce CLINIC-GENIE, an explainable two-stage framework consisting of: (1) CLINIC, a dysarthria severity classification model combining acoustic and speech embeddings with Clinically Explainable Acoustic Features (CEAFs); and (2) GENIE, a module translating CEAFs and their Shapley values into intuitive natural language explanations via a large language model. CLINIC achieved a balanced accuracy of 0.952 (17.3% improvement over using CEAFs alone), and certified speech-language pathologists rated explanations from CLINIC-GENIE with an average fidelity score of 4.94, confirming enhanced clinical utility.