Graphs are ubiquitous in social networks, chemical molecules, and financial data, where Graph Neural Networks (GNNs) achieve superior predictive accuracy. Graphs can beevolving, while understanding how GNN predictions respond to the evolution provides significant insight and trust. We explore the problem of explaining evolving GNN predictions due to continuously changing edge weights.We introduce a layer edge-based explanation to balanceexplanation fidelity and interpretability.We propose a novel framework to address the challenges of axiomatic attribution and the entanglement of multiple computational graph paths due to continuous change of edge weights. We first design an axiomatic attribution of the evolution of the model prediction to message flows, then develop Shapley value to fairly map message flow contributions to layer edges.We formulate a novel optimization problem to find the critical layer edges based on KL-divergence minimization. Extensive experiments on eight datasets for node classification, link prediction, and graph classification tasks with evolving graphs demonstrate the better fidelity and interpretability of the proposed method over the baseline methods. The code is available at https://github.com/yazhengliu/Axiomatic-Layer-Edges/tree/main.