Amidst the surge in the use of Artificial Intelligence (AI) for control purposes, classical and model-based control methods maintain their popularity due to their transparency and deterministic nature. However, advanced controllers like Nonlinear Model Predictive Control (NMPC), despite proven capabilities, face adoption challenges due to their computational complexity and unpredictable closed-loop performance in complex validation systems. This paper introduces ExAMPC, a methodology bridging classical control and explainable AI by augmenting the NMPC with data-driven insights to improve the trustworthiness and reveal the optimization solution and closed-loop performance's sensitivities to physical variables and system parameters. By employing a low-order spline embedding, we reduce the open-loop trajectory dimensionality by over 95%, and integrate it with SHAP and Symbolic Regression from eXplainable AI (XAI) for an approximate NMPC, enabling intuitive physical insights into the NMPC's optimization routine. The prediction accuracy of the approximate NMPC is enhanced through physics-inspired continuous-time constraints penalties, reducing the predicted continuous trajectory violations by 93%. ExAMPC also enables accurate forecasting of the NMPC's computational requirements with explainable insights on worst-case scenarios. Experimental validation on automated valet parking and autonomous racing with lap-time optimization, demonstrates the methodology's practical effectiveness for potential real-world applications.