Human activity recognition (HAR) has become a key component of intelligent systems for healthcare monitoring, assistive living, smart environments, and human-computer interaction. Although deep learning has substantially improved HAR performance on multivariate sensor data, the resulting models often remain opaque, limiting trust, reliability, and real-world deployment. Explainable artificial intelligence (XAI) has therefore emerged as a critical direction for making HAR systems more transparent and human-centered. This paper presents a comprehensive review of explainable HAR methods across wearable, ambient, physiological, and multimodal sensing settings. We introduce a unified perspective that separates conceptual dimensions of explainability from algorithmic explanation mechanisms, reducing ambiguities in prior surveys. Building on this distinction, we present a mechanism-centric taxonomy of XAI-HAR methods covering major explanation paradigms. The review examines how these methods address the temporal, multimodal, and semantic complexities of HAR, and summarize their interpretability objectives, explanation targets, and limitations. In addition, we discuss current evaluation practices, highlight key challenges in achieving reliable and deployable XAI-HAR, and outline directions toward trustworthy activity recognition systems that better support human understanding and decision-making.