Large language models (LLMs) excel when user goals are clearly specified, yet real-world queries are often vague and evolving, forcing LLMs to guess and leading to misaligned responses. Existing approaches attempt to clarify user intents through iterative questioning. While effective in alleviating disambiguation, this paradigm tends to merely provide standard and normal responses, which fails to meet the growing demand for diverse personalized expression of users. Based on the observation, we first identify its problem as the ignore of users' mental states, in which the complicated mental elements, unclear functional rules, and evolving mental states pose obstacles to approach the problem. Therefore, in this paper, we introduce the theory of the mentalizing network in the human brain and propose a neurobiologically inspired framework, i.e., NeurPIU, that endows LLMs with human-like mentalizing capabilities for personalized intent understanding. NeurPIU constructs an intent neural network that organizes users' long-term mental states into a three-layer graph; retrieves query-relevant states via spreading activation mechanism with temporal decay; and injects an encoded cognitive prefix into a frozen LLM through a lightweight LoRA-based cognitive model to guide response generation. The network is incrementally updated after each interaction to track evolving user cognition. Extensive experiments on four benchmarks show that NeurPIU consistently improves long-term dialogue quality, especially its plug-and-play feature, and generalizes to conversational recommendation and mental health counseling. A user study with physiological measurements further indicates that NeurPIU reduces interaction time by 53.95% while improving user experience ratings by 22.38%. All data and code are released.