Robotic grasping is a fundamental ability for a robot to interact with the environment. Current methods focus on how to obtain a stable and reliable grasping pose in object level, while little work has been studied on part (shape)-wise grasping which is related to fine-grained grasping and robotic affordance. Parts can be seen as atomic elements to compose an object, which contains rich semantic knowledge and a strong correlation with affordance. However, lacking a large part-wise 3D robotic dataset limits the development of part representation learning and downstream applications. In this paper, we propose a new large Language-guided SHape grAsPing datasEt (named LangSHAPE) to promote 3D part-level affordance and grasping ability learning. From the perspective of robotic cognition, we design a two-stage fine-grained robotic grasping framework (named LangPartGPD), including a novel 3D part language grounding model and a part-aware grasp pose detection model, in which explicit language input from human or large language models (LLMs) could guide a robot to generate part-level 6-DoF grasping pose with textual explanation. Our method combines the advantages of human-robot collaboration and LLMs' planning ability using explicit language as a symbolic intermediate. To evaluate the effectiveness of our proposed method, we perform 3D part grounding and fine-grained grasp detection experiments on both simulation and physical robot settings, following language instructions across different degrees of textual complexity. Results show our method achieves competitive performance in 3D geometry fine-grained grounding, object affordance inference, and 3D part-aware grasping tasks. Our dataset and code are available on our project website https://sites.google.com/view/lang-shape