Explanations of machine learning (ML) model predictions generated by Explainable AI (XAI) techniques such as SHAP are essential for people using ML outputs for decision-making. We explore the potential of Large Language Models (LLMs) to transform these explanations into human-readable, narrative formats that align with natural communication. We address two key research questions: (1) Can LLMs reliably transform traditional explanations into high-quality narratives? and (2) How can we effectively evaluate the quality of narrative explanations? To answer these questions, we introduce Explingo, which consists of two LLM-based subsystems, a Narrator and Grader. The Narrator takes in ML explanations and transforms them into natural-language descriptions. The Grader scores these narratives on a set of metrics including accuracy, completeness, fluency, and conciseness. Our experiments demonstrate that LLMs can generate high-quality narratives that achieve high scores across all metrics, particularly when guided by a small number of human-labeled and bootstrapped examples. We also identified areas that remain challenging, in particular for effectively scoring narratives in complex domains. The findings from this work have been integrated into an open-source tool that makes narrative explanations available for further applications.