Despite the success of large language models (LLMs) across domains, their potential for efficient channel state information (CSI) compression and feedback in frequency division duplex (FDD) massive multiple-input multiple-output (mMIMO) systems remains largely unexplored yet increasingly important. In this paper, we propose a novel LLM-based framework for CSI feedback to exploit the potential of LLMs. We first reformulate the CSI compression feedback task as a masked token prediction task that aligns more closely with the functionality of LLMs. Subsequently, we design an information-theoretic mask selection strategy based on self-information, identifying and selecting CSI elements with the highest self-information at the user equipment (UE) for feedback. This ensures that masked tokens correspond to elements with lower self-information, while visible tokens correspond to elements with higher self-information, thus maximizing the accuracy of LLM predictions. Finally, the LLM leverages its robust modeling capabilities to reconstruct complete CSI representations through contextual inference. This self-information-driven masking strategy integrates the LLM-based masked token prediction mechanism into a coherent, information-driven framework. Numerical results indicate that the proposed LLM-based CSI feedback framework significantly outperforms traditional small models in CSI reconstruction accuracy, leading to substantial improvements in communication rates in multi-user MIMO scenarios. This approach has the potential to address the limitations of CSI reconstruction accuracy that restrict multi-user communication rates. Moreover, the method deploys a lightweight network at the UE, with additional network complexity overhead only at the base station (BS). Finally, the method demonstrates strong generalization across different compression ratios and exhibits excellent transfer learning capabilities across various channel scenarios. These findings pave the way for integrating LLMs into next-generation wireless communication systems.