Lower limb amputation severely impairs mobility, and current prosthetic devices ranging from passive mechanical limbs to microprocessor-controlled systems fail to provide intuitive, adaptive control that restores natural gait. A critical barrier is the inability of existing control systems to interpret and adapt to individual users' neuromuscular signals in real time, leading to high cognitive load, gait asymmetry, and elevated fall risk. This paper presents XAI-Prosthesis, an explainable artificial intelligence framework for real-time, adaptive prosthetic control that integrates surface electromyography (sEMG) and inertial measurement unit (IMU) signals from the residual limb to predict user intent and continuously optimize prosthetic joint (knee and ankle) trajectories. The framework comprises three core components: (1) a lightweight temporal convolutional network (TCN) with attention that extracts both transient (intent onset) and sustained (gait phase) features from 8-channel sEMG and 6-axis IMU data at 200 Hz; (2) a real-time adaptation module employing online Bayesian inference that updates model parameters every 100 ms based on error between predicted and actual prosthetic motion, accommodating neuromuscular fatigue, muscle activation pattern drift, and walking terrain changes (level ground, stairs, ramps, uneven terrain); and (3) an explainability layer using SHAP (SHapley Additive exPlanations) and counterfactual explanations that provides users and clinicians with interpretable feedback on why specific control decisions were made, enhancing trust and enabling informed clinical tuning. We validate XAI-Prosthesis on a longitudinal dataset from 24 unilateral transtibial amputees (mean age 48.2 ± 12.4 years, 38% female) who used the system during 6 months of home and community ambulation (total 8,400 hours of walking data). The system achieves intent recognition accuracy of 95.7% for five locomotion modes (level walking, stair ascent/descent, ramp ascent/descent) with average decision latency of 48 ms substantially faster than existing myoelectric control systems (typically 150-250 ms). Real-time adaptation reduces prosthetic joint angle trajectory error from 3.8° to 1.2° RMS compared to non-adaptive baseline (68% reduction). Gait symmetry (symmetry index) improves from 0.21 to 0.09 (57% improvement), and user-reported prosthetic satisfaction (PQ-Score) increases from 58 to 84 out of 100. Explainability feedback enables clinicians to identify problematic neuromuscular patterns and adjust prosthetic parameters without removing the device. Key neuromuscular predictors of gait intent include: integrated EMG of gastrocnemius lateralis (35% predictive weight), co-contraction index of tibialis anterior and soleus (28%), and IMU-derived shank angular velocity (22%). The model runs on an embedded edge device (NVIDIA Jetson Orin Nano, 10W) with inference latency of 12 ms per time step, supporting real-time control. This work establishes that explainable AI, combining deep learning with online adaptation and interpretability, can transform prosthetic control from rigid, user-dependent calibration to personalized, adaptive, and trustworthy human-machine collaboration.