Graph Neural Networks (GNNs) are now a standard tool for modelling graphstructured data in applications such as molecular property prediction, drug discovery, recommender systems, and citation networks. In high-stakes domains, especially computational biology and chemoinformatics, practitioners increasingly rely on GNNs for decision-making on relational data. However, despite their strong predictive performance, they still face several important challenges. It is still difficult to understand how GNNs organise information across layers, how depth affects the quality of the learned representations, and how individual decisions can be explained in human terms. The work in this thesis is driven by the central idea that combining activation-centred analysis with rule-based explanations can be the key to more transparent decision-making. The first challenge concerns the explainability of GNNs. As with other neural network-based models, they suffer from the black-box problem, making it difficult to understand the processes that drive their predictions. In this thesis, we address part of this challenge by focusing on graph convolutional network architecture on benchmark graph datasets, primarily in node classification settings. We introduce Functional Semantic Activation Mapping, a representation that acts as an intermediate layer between the knowledge encoded in a trained GNN and a human-interpretable semantic structure. Given a trained GNN, we first show how a semantic activation graph can be constructed and used to obtain global insights into the organisation of classes, the appearance and loss of class-specific structure with depth, and the relationship between activation communities and observed patterns of class confusion. This analysis is then extended to examine depth, oversmoothing, and their effect on discriminative power. Building on the same activation extraction method, we introduce a semantic activation-mining approach that extracts compact rule-based explanations from binarised hidden activations, producing small sets of activation rules together with local subgraph rationales for individual predictions. Finally, the thesis outlines several research directions for neurosymbolic explainability in graph-based learning systems.