Recent improvements in the expressive power of spatio-temporal models have led to performance gains in many real-world applications, such as traffic forecasting and social network modelling. However, understanding the predictions from a model is crucial to ensure reliability and trustworthiness, particularly for high-risk applications, such as healthcare and transport. Few existing methods are able to generate explanations for models trained on continuous-time dynamic graph data and, of these, the computational complexity and lack of suitable explanation objectives pose challenges. In this paper, we propose $\textbf{S}$patio-$\textbf{T}$emporal E$\textbf{X}$planation $\textbf{Search}$ (STX-Search), a novel method for generating instance-level explanations that is applicable to static and dynamic temporal graph structures. We introduce a novel search strategy and objective function, to find explanations that are highly faithful and interpretable. When compared with existing methods, STX-Search produces explanations of higher fidelity whilst optimising explanation size to maintain interpretability.