For the intuition's sake, perhaps it would be useful to visualize the ripples produced by throwing a stone into a pond. Close to where the stone impacted the pond's surface, the wave fronts are strongly curved, and are not approximated well by a plane wave. However, at large distances from the point of impact the curvature of the wavefronts is very small, and a plane wave approximation is more accurate.

Because the wave equation for electromagnetic waves in free space is essentially identical to that governing the surface waves on a pond, the above picture generalizes to give a picture of light waves produced by, say, an oscillating dipole. When you are sufficiently far from the dipole, the radiation field will be approximately plane.