Continuum robots possess high flexibility and redundancy, making them well suited for safe interaction in complex environments, yet their continuous deformation and nonlinear dynamics pose fundamental challenges to perception, modeling, and control. Existing vision-based control approaches often rely on end-to-end learning, achieving shape regulation without explicit awareness of robot geometry or its interaction with the environment. Here, we introduce a shape-interpretable visual self-modeling framework for continuum robots that enables geometry-aware control. Robot shapes are encoded from multi-view planar images using a Bezier-curve representation, transforming visual observations into a compact and physically meaningful shape space that uniquely characterizes the robot's three-dimensional configuration. Based on this representation, neural ordinary differential equations are employed to self-model both shape and end-effector dynamics directly from data, enabling hybrid shape-position control without analytical models or dense body markers. The explicit geometric structure of the learned shape space allows the robot to reason about its body and surroundings, supporting environment-aware behaviors such as obstacle avoidance and self-motion while maintaining end-effector objectives. Experiments on a cable-driven continuum robot demonstrate accurate shape-position regulation and tracking, with shape errors within 1.56% of image resolution and end-effector errors within 2% of robot length, as well as robust performance in constrained environments. By elevating visual shape representations from two-dimensional observations to an interpretable three-dimensional self-model, this work establishes a principled alternative to vision-based end-to-end control and advances autonomous, geometry-aware manipulation for continuum robots.