Accurate modeling of car-following behaviors is crucial for autonomous driving systems. While recent advancements in large language models (LLMs) have shown promise in various domains, their application to car-following tasks has not been fully explored. In this study, we introduce GenFollower, a novel framework that leverages LLMs for car-following modeling. GenFollower employs a specialized prompt engineering technique that integrates heterogeneous inputs into structured natural language prompts, incorporating Chain of Thought (CoT) to enhance reasoning and clarity. GenFollower prioritizes interpretability by explicitly requiring explanations into the prompts, providing valuable insights into the model's decision-making process. Through experiments on the Waymo Open datasets, we demonstrate GenFollower's superior performance and interpretability. Our findings contribute to advancing the field of car-following modeling and pave the way for more robust and reliable autonomous driving systems.