The performance of modern DBMSs such as MySQL and PostgreSQL heavily depends on the configuration of performance-critical knobs. Manual tuning these knobs is laborious and inefficient due to the complex and high-dimensional nature of the configuration space. Among the automated tuning methods, reinforcement learning (RL)-based methods have recently sought to improve the DBMS knobs tuning process from several different perspectives. However, they still encounter challenges with slow convergence speed during offline training. In this paper, we mainly focus on how to leverage the valuable tuning hints contained in various textual documents such as DBMS manuals and web forums to improve the offline training of RL-based methods. To this end, we propose an efficient DBMS knobs tuning framework named DemoTuner via a novel LLM-assisted demonstration reinforcement learning method. Specifically, to comprehensively and accurately mine tuning hints from documents, we design a structured chain of thought prompt to employ LLMs to conduct a condition-aware tuning hints extraction task. To effectively integrate the mined tuning hints into RL agent training, we propose a hint-aware demonstration reinforcement learning algorithm HA-DDPGfD in DemoTuner. As far as we know, DemoTuner is the first work to introduce the demonstration reinforcement learning algorithm for DBMS knobs tuning. Experimental evaluations conducted on MySQL and PostgreSQL across various workloads demonstrate that DemoTuner achieves performance gains of up to 44.01% for MySQL and 39.95% for PostgreSQL over default configurations. Compared with three representative baseline methods, DemoTuner is able to further reduce the execution time by up to 10.03%, while always consuming the least online tuning cost. Additionally, DemoTuner also exhibits superior adaptability to application scenarios with unknown workloads.