Why do plano-convex lenses reduce spherical aberation?

Plano-convex lenses only reduce the spherical aberration if they are used in the appropriate position with respect to the conjugates. To understand this, one intuitive explanation is this: Snell's law is not linear, so it does not behave proportionately for large versus small angles.

Looking at the two lenses below, for the left hand lens the angles of deviation for the marginal ray (the ray at the edge of the lens) are distributed between the front and back surfaces. For the right hand lens, the 2nd surface performs all of the work of refraction. It turns out the right hand lens will have much more spherical aberration than the left hand lens, when used in this manner.

This is a simplified example, but more complicated lenses are designed by adjusting the amount of refraction at each surface, ideally so that aberrations on one surface are compensated by aberrations on another surface.