Smart agricultural harvesting robots’ vision recognition, control decision, and mechanical hand modules all resemble the human eye, brain, and hand, respectively. To enable automatic and precise picking of target fruits and vegetables, the system makes use of cutting-edge sensor technology, machine vision algorithms, and intelligent control and decision methods. This paper provides a comprehensive review of international research advancements in the “eye–brain–hand” harvesting systems within the context of smart agriculture, encompassing aspects of mechanical hand devices, visual recognition systems, and intelligent decision systems. Then, the key technologies used in the current research are reviewed, including image processing, object detection and tracking, machine learning, deep learning, etc. In addition, this paper explores the application of the system to different crops and environmental conditions and analyzes its advantages and challenges. Finally, the challenges and prospects for the research on picking robots in the future are presented, including further optimization of the algorithm and improvement of flexibility and reliability of mechanical devices. To sum up, the “eye–brain–hand” picking system in intelligent agriculture has great potential to improve the efficiency and quality of crop picking and reduce labor pressure, and it is expected to be widely used in agricultural production.