Evacuation decision prediction is critical for efficient and effective wildfire response by helping emergency management anticipate traffic congestion and bottlenecks, allocate resources, and minimize negative impacts. Traditional statistical methods for evacuation decision prediction fail to capture the complex and diverse behavioral logic of different individuals. In this work, for the first time, we introduce FLARE, short for facilitating LLM for advanced reasoning on wildfire evacuation decision prediction, a Large Language Model (LLM)-based framework that integrates behavioral theories and models to streamline the Chain-of-Thought (CoT) reasoning and subsequently integrate with memory-based Reinforcement Learning (RL) module to provide accurate evacuation decision prediction and understanding. Our proposed method addresses the limitations of using existing LLMs for evacuation behavioral predictions, such as limited survey data, mismatching with behavioral theory, conflicting individual preferences, implicit and complex mental states, and intractable mental state-behavior mapping. Experiments on three post-wildfire survey datasets show an average of 20.47% performance improvement over traditional theory-informed behavioral models, with strong cross-event generalizability. Our complete code is publicly available at https://github.com/SusuXu-s-Lab/FLARE