An aerial manipulator, comprising a multirotor base and a robotic arm, is subject to significant dynamic coupling between these two components. Therefore, achieving precise and robust motion control is a challenging yet important objective. Here, we propose a novel prescribed performance motion control framework based on variable-gain extended state observers (ESOs), referred to as PreGME. The method includes variable-gain ESOs for real-time estimation of dynamic coupling and a prescribed performance flight control that incorporates error trajectory constraints. Compared with existing methods, the proposed approach exhibits the following two characteristics. First, the adopted variable-gain ESOs can accurately estimate rapidly varying dynamic coupling. This enables the proposed method to handle manipulation tasks that require aggressive motion of the robotic arm. Second, by prescribing the performance, a preset error trajectory is generated to guide the system evolution along this trajectory. This strategy allows the proposed method to ensure the tracking error remains within the prescribed performance envelope, thereby achieving high-precision control. Experiments on a real platform, including aerial staff twirling, aerial mixology, and aerial cart-pulling experiments, are conducted to validate the effectiveness of the proposed method. Experimental results demonstrate that even under the dynamic coupling caused by rapid robotic arm motion (end-effector velocity: 1.02 m/s, acceleration: 5.10 m/s$^2$), the proposed method achieves high tracking performance.