An explanation of this is put forth in both the video and paper linked to in the opening paragraph. The principle is that bends in rivers tend to grow as erosion happens on the outside of the bend and soil deposition on the inside. This increases sinuosity until the point at which the bend comes full circle, forms an oxbox lake, and returns the local region of the river to a straight line with sinuosity of 1. The value of pi is supposed to come out when you consider all of a river's curves and wiggles on all length scales.

The right answer may not be pi, but the data shown make a compelling case that rivers do tend to some average value.

As a sidenote, there are active human efforts to keep certain rivers, like the Mississippi, from meandering too far from their current locations. I don't know how many of the world's rivers have such efforts being applied to them, but it's not unreasonable to think that this could have some effect.