Quadrupedal jumping has been intensively investigated in recent years. Still, realizing controlled jumping with soft landings remains an open challenge due to the complexity of the jump dynamics and the need to perform complex computations during the short time. This work tackles this challenge by leveraging trajectory optimization and behavior cloning. We generate an optimal jumping motion by utilizing dual-layered coarse-to-refine trajectory optimization. We combine this with a variable impedance control approach to achieve soft landing. Finally, we distill this computationally heavy jumping and landing policy into an efficient neural network via behavior cloning. Extensive simulation experiments demonstrate that, compared to classic model predictive control, the variable impedance control ensures compliance and reduces the stress on the motors during the landing phase. Furthermore, the neural network can reproduce jumping and landing behavior, achieving at least a 97.4% success rate. Hardware experiments confirm the findings, showcasing explosive jumping with soft landings and on-the-fly evaluation of the control actions.