The idea is that since the measurements of the gravitational constant G haven't all been the same, perhaps one can find that G is a function of time. The plot that tells it all is below:
The first reaction should be "Ack!" As somebody pointed out today, if you take just about any set of points you can fit them with a sinusoid if you make the frequency high enough. And, of course, the points represent measurements taken over a non-trivial length of time that may not be closely related to the publication date.
But Lorenzo Iorio decided to take the question seriously. He predicted the orbit of Saturn (no, I don't know why Saturn, maybe he had the data handy) and compared what you expect if G was constant with what you should get if G varied the way the first paper intimated. With G constant he can extrapolate from older observations and predict the position of Saturn to 0.1km. If G varied as mentioned above, the displacement would be O(100,000km). Good for him. Raspberries for JD Anderson et al.
FWIW, it isn't insane to think that G might be changing, but I'd expect an exponential or an inverse power of time, not a wiggle. "Cooper has a stream which is fifty feet wide where it flows out of a lake; it presently narrows to twenty as it meanders along for no given reason, and yet when a stream acts like that it ought to be required to explain itself."