Controlling soft robots with precision is a challenge due in large part to the difficulty of constructing models that are amenable to model-based control design techniques. Koopman Operator Theory offers a way to construct explicit linear dynamical models of soft robots and to control them using established model-based linear control methods. This method is data-driven, yet unlike other data-driven models such as neural networks, it yields an explicit control-oriented linear model rather than just a "black-box" input-output mapping. This work describes this Koopman-based system identification method and its application to model predictive controller design. A model and MPC controller of a pneumatic soft robot arm was constructed via the method, and its performance was evaluated over several trajectory following tasks in the real-world. On all of the tasks, the Koopman-based MPC controller outperformed a benchmark MPC controller based on a linear state-space model of the same system.