A decent explanation is here:
http://www.physicspages.com/2013/03/30/geodesics-paths-of-lo...
(I haven't checked all the calculations though).
A similar explanation is shown in Feynman's lectures:http://www.feynmanlectures.caltech.edu/II_42.html
Finally, an explanation of gravity in layman's terms.
Still, there is the kernel of the correct explanation which could probably be presented to the larger public, starting with the well known explanation for refraction in terms of minimal paths (the lifeguard trying to save a drowning man), passing to the fact that in special relativity the trajectory of the free particle maximizes the proper time, and ending up with the gravitational case.