Graph machine learning (GML) has been successfully applied across a wide range of tasks. Nonetheless, GML faces significant challenges in generalizing over out-of-distribution (OOD) data, which raises concerns about its wider applicability. Recent advancements have underscored the crucial role of causality-driven approaches in overcoming these generalization challenges. Distinct from traditional GML methods that primarily rely on statistical dependencies, causality-focused strategies delve into the underlying causal mechanisms of data generation and model prediction, thus significantly improving the generalization of GML across different environments. This paper offers a thorough review of recent progress in causality-involved GML generalization. We elucidate the fundamental concepts of employing causality to enhance graph model generalization and categorize the various approaches, providing detailed descriptions of their methodologies and the connections among them. Furthermore, we explore the incorporation of causality in other related important areas of trustworthy GML, such as explanation, fairness, and robustness. Concluding with a discussion on potential future research directions, this review seeks to articulate the continuing development and future potential of causality in enhancing the trustworthiness of graph machine learning.