Soft robots offer unmatched adaptability and safety in unstructured environments, yet their compliant, high-dimensional, and nonlinear dynamics make modeling for control notoriously difficult. Existing data-driven approaches often fail to generalize, constrained by narrowly focused task demonstrations or inefficient random exploration. We introduce SoftAE, an uncertainty-aware active exploration framework that autonomously learns task-agnostic and generalizable dynamics models of soft robotic systems. SoftAE employs probabilistic ensemble models to estimate epistemic uncertainty and actively guides exploration toward underrepresented regions of the state-action space, achieving efficient coverage of diverse behaviors without task-specific supervision. We evaluate SoftAE on three simulated soft robotic platforms -- a continuum arm, an articulated fish in fluid, and a musculoskeletal leg with hybrid actuation -- and on a pneumatically actuated continuum soft arm in the real world. Compared with random exploration and task-specific model-based reinforcement learning, SoftAE produces more accurate dynamics models, enables superior zero-shot control on unseen tasks, and maintains robustness under sensing noise, actuation delays, and nonlinear material effects. These results demonstrate that uncertainty-driven active exploration can yield scalable, reusable dynamics models across diverse soft robotic morphologies, representing a step toward more autonomous, adaptable, and data-efficient control in compliant robots.