In order to enhance the amphibious mobility of robots in water-land environments, this paper proposes a frog-inspired hybrid drive amphibious robot, based on the amphibious locomotion characteristics of frogs. Distinct from existing single-mode frog-inspired jumping or swimming robots, the proposed robot innovatively integrates hybrid propulsion to simultaneously achieve both frog-like swimming and jumping capabilities. On land, the robot utilizes an explosion-driven hind limb actuation mechanism, paired with a linkage-based forelimb posture control system, to achieve high-performance frog-like jumping. In water, a rope-driven hind limb mechanism facilitates extension and retraction movements, while controllable soft-actuated flippers enable swinging and opening/closing motions, thereby achieving efficient frog-like swimming. In addition, an amphibious dynamic model was developed, and the robot's amphibious locomotion capabilities were evaluated and analyzed. Finally, an experimental prototype platform was built to test the amphibious locomotion performance of the designed robot, and a comparative analysis was conducted with the theoretical model. The experimental results not only validated the correctness of the amphibious dynamics and motion theory, but also confirmed the effectiveness of the designed amphibious terrain-crossing mechanism.