Personal development through self-directed learning is essential in today's fast-changing world, but many learners struggle to manage it effectively. While AI tools like large language models (LLMs) have the potential for personalized learning planning, they face issues such as transparency and hallucinated information. To address this, we propose PlanGlow, an LLM-based system that generates personalized, well-structured study plans with clear explanations and controllability through user-centered interactions. Through mixed methods, we surveyed 28 participants and interviewed 10 before development, followed by a within-subject experiment with 24 participants to evaluate PlanGlow's performance, usability, controllability, and explainability against two baseline systems: a GPT-4o-based system and Khan Academy's Khanmigo. Results demonstrate that PlanGlow significantly improves usability, explainability, and controllability. Additionally, two educational experts assessed and confirmed the quality of the generated study plans. These findings highlight PlanGlow's potential to enhance personalized learning and address key challenges in self-directed learning.