To enable robots to use tools, the initial step is teaching robots to employ dexterous gestures for touching specific areas precisely where tasks are performed. Affordance features of objects serve as a bridge in the functional interaction between agents and objects. However, leveraging these affordance cues to help robots achieve functional tool grasping remains unresolved. To address this, we propose a granularity-aware affordance feature extraction method for locating functional affordance areas and predicting dexterous coarse gestures. We study the intrinsic mechanisms of human tool use. On one hand, we use fine-grained affordance features of object-functional finger contact areas to locate functional affordance regions. On the other hand, we use highly activated coarse-grained affordance features in hand-object interaction regions to predict grasp gestures. Additionally, we introduce a model-based post-processing module that transforms affordance localization and gesture prediction into executable robotic actions. This forms GAAF-Dex, a complete framework that learns Granularity-Aware Affordances from human-object interaction to enable tool-based functional grasping with dexterous hands. Unlike fully-supervised methods that require extensive data annotation, we employ a weakly supervised approach to extract relevant cues from exocentric (Exo) images of hand-object interactions to supervise feature extraction in egocentric (Ego) images. To support this approach, we have constructed a small-scale dataset, Functional Affordance Hand-object Interaction Dataset (FAH), which includes nearly 6K images of functional hand-object interaction Exo images and Ego images. Extensive experiments on the dataset demonstrate that our method outperforms state-of-the-art methods. The source code and the established dataset are available at https://github.com/yangfan293/GAAF-DEX.