Knowledge graph completion (KGC) aims to alleviate the inherent incompleteness of knowledge graphs (KGs), which is a critical task for various applications, such as recommendations on the web. Although knowledge graph embedding (KGE) models have demonstrated superior predictive performance on KGC tasks, these models infer missing links in a black-box manner that lacks transparency and accountability, preventing researchers from developing accountable models. Existing KGE-based explanation methods focus on exploring key paths or isolated edges as explanations, which is information-less to reason target prediction. Additionally, the missing ground truth leads to these explanation methods being ineffective in quantitatively evaluating explored explanations. To overcome these limitations, we propose KGExplainer, a model-agnostic method that identifies connected subgraph explanations and distills an evaluator to assess them quantitatively. KGExplainer employs a perturbation-based greedy search algorithm to find key connected subgraphs as explanations within the local structure of target predictions. To evaluate the quality of the explored explanations, KGExplainer distills an evaluator from the target KGE model. By forwarding the explanations to the evaluator, our method can examine the fidelity of them. Extensive experiments on benchmark datasets demonstrate that KGExplainer yields promising improvement and achieves an optimal ratio of 83.3% in human evaluation.