This paper presents a novel biomimetic morphing quadrotor design inspired by the morphology of an eagle claw during prey capture. The arms of the quadrotor are capable of vertical folding to enable dynamic grasping, mimicking the transition of the eagle claw from an open to a closed state. This transition is achieved through the rotation of a central servomotor and the associated movement of 20 links. Thanks to the closed-loop multi-link structure of the frame, the propellers of the quadrotor remain in a fixed orientation when the arms are folded, allowing for system stabilization at any arm rotation angle. The geometric property of the whole frame is analyzed to determine the relationships and constraints of the links, which is important in experimental vehicle fabrication. To handle possible physical property changes and external disturbances during grasping, the adaptive sliding mode controllers are applied. To deal with objects of unknown size in grasping tasks, an admittance filter is proposed for adaptive morphology. While in flight, our proposed morphing quadrotor is able to rapidly or continuously transition to any configuration within its range smoothly. Experimental results show the ability of the quadrotor to dynamically grasp various unknown objects at 0.4m/s without additional tools, as well as its versatility in traversal of narrow spaces and perching.