The moment of inertia relates to resisting rotational acceleration, which involves motion. With respect to a rotational influence twisting at the axis, the radius affects the amount of this resistance two separate (although intertwined) ways. Firstly, if the radius is doubled then the mechanical advantage of the lever arm (radius) is reduced by a factor of two, it will require twice the torque at the axis to produce the same force out at the end of the lever arm. Secondly, doubling the radius doubles the length of the arc sweep when motion occurs. For any arbitrary bit of rotation at the axis, doubling the radius will double the distance along the arc swept by the end of the lever arm (radius), necessitating the tip of the lever arm to travel at twice the velocity.
So an increased radius participates in resisting rotational acceleration two ways, accounting for the appearance of the $r^2$ rather than simply $r$.
Put another way, when you double the radius, for any given twist applied at the axis, you are trying to accelerate the mass twice as quickly with only half as much leverage.
