Large Language Models (LLMs) based agents have demonstrated remarkable potential in autonomous task-solving across complex, open-ended environments. A promising approach for improving the reasoning capabilities of LLM agents is to better utilize prior experiences in guiding current decisions. However, LLMs acquire experience either through implicit memory via training, which suffers from catastrophic forgetting and limited interpretability, or explicit memory via prompting, which lacks adaptability. In this paper, we introduce a novel agent-centric, trainable, multi-layered graph memory framework and evaluate how context memory enhances the ability of LLMs to utilize parametric information. The graph abstracts raw agent trajectories into structured decision paths in a state machine and further distills them into high-level, human-interpretable strategic meta-cognition. In order to make memory adaptable, we propose a reinforcement-based weight optimization procedure that estimates the empirical utility of each meta-cognition based on reward feedback from downstream tasks. These optimized strategies are then dynamically integrated into the LLM agent's training loop through meta-cognitive prompting. Empirically, the learnable graph memory delivers robust generalization, improves LLM agents' strategic reasoning performance, and provides consistent benefits during Reinforcement Learning (RL) training.