Underactuated soft robot hands offer inherent safety and adaptability advantages over rigid systems, but developing dexterous manipulation skills remains challenging. While imitation learning shows promise for complex manipulation tasks, traditional approaches struggle with soft systems due to demonstration collection challenges and ineffective state representations. We present KineSoft, a framework enabling direct kinesthetic teaching of soft robotic hands by leveraging their natural compliance as a skill teaching advantage rather than only as a control challenge. KineSoft makes two key contributions: (1) an internal strain sensing array providing occlusion-free proprioceptive shape estimation, and (2) a shape-based imitation learning framework that uses proprioceptive feedback with a low-level shape-conditioned controller to ground diffusion-based policies. This enables human demonstrators to physically guide the robot while the system learns to associate proprioceptive patterns with successful manipulation strategies. We validate KineSoft through physical experiments, demonstrating superior shape estimation accuracy compared to baseline methods, precise shape-trajectory tracking, and higher task success rates compared to baseline imitation learning approaches.