Artificial intelligence (AI)-driven augmented reality (AR) systems are becoming increasingly integrated into daily life, and with this growth comes a greater need for explainability in real-time user interactions. Traditional explainable AI (XAI) methods, which often rely on feature-based or example-based explanations, struggle to deliver dynamic, context-specific, personalized, and human-centric insights for everyday AR users. These methods typically address separate explainability dimensions (e.g., when, what, how) with different explanation techniques, resulting in unrealistic and fragmented experiences for seamless AR interactions. To address this challenge, we propose PILAR, a novel framework that leverages a pre-trained large language model (LLM) to generate context-aware, personalized explanations, offering a more intuitive and trustworthy experience in real-time AI-powered AR systems. Unlike traditional methods, which rely on multiple techniques for different aspects of explanation, PILAR employs a unified LLM-based approach that dynamically adapts explanations to the user's needs, fostering greater trust and engagement. We implement the PILAR concept in a real-world AR application (e.g., personalized recipe recommendations), an open-source prototype that integrates real-time object detection, recipe recommendation, and LLM-based personalized explanations of the recommended recipes based on users' dietary preferences. We evaluate the effectiveness of PILAR through a user study with 16 participants performing AR-based recipe recommendation tasks, comparing an LLM-based explanation interface to a traditional template-based one. Results show that the LLM-based interface significantly enhances user performance and experience, with participants completing tasks 40% faster and reporting greater satisfaction, ease of use, and perceived transparency.