I haven't seen this particular equation, probably because according to the
Wikipedia article, the results have not been duplicated (making them effectively discredited).
But anyway, it's not that hard to understand, especially if you go back to the original paper from 1986. Basically, the authors of that paper (Fischbach et al.) were looking at some experimental measurements of the gravitational constant, G. They noticed that geophysical measurements of G, i.e. measured using variations in gravity around the Earth, were giving slightly larger numbers than the "laboratory value" of G which is based on small-scale measurements using an
Eötvös torsion balance and the like. The point is that the torsion balance measurements involved objects which were not as far apart as the objects in the geophysical measurements, e.g. two spheres in a lab are not as far apart as the Earth and a satellite, so if the force between two massive objects depends on their separation in some way other than F \propto 1/r^2, it would show up as a discrepancy in the value of G between these two different classes of experiments.
Now, the normal gravitational potential energy of two masses is given by
V_g(r) = -\frac{Gm_1 m_2}{r}
If there is another force at work, because of
calculations from quantum field theory, we would expect it to make an additional contribution to the potential energy of the form
V_5(r) = -Gm_1 m_2\frac{\alpha e^{-r/\lambda}}{r}
The constant \lambda can be loosely interpreted as the "range" of the force, and \alpha is related to the strength of the force. So the total potential, corresponding to gravity plus this "fifth force," would be
V(r) = -\frac{Gm_1 m_2}{r}(1 + \alpha e^{-r\lambda})
The authors of the 1986 paper checked the experimental data to see whether it was consistent with a potential energy of this form, and they concluded that it was. By performing a fit to the data (I presume) they identified the specific values of \alpha and \lambda that best matched the measurements, namely \alpha = (-7.2\pm 3.6)\times 10^{-3} and \lambda = (200 \pm 50)\ \mathrm{m}.
The excerpt from the textbook talks about the force that would correspond to this potential energy, which you can derive using the usual definition F = -\partial V/\partial r. Basically it is showing you mathematically that, at large distances (much larger than the "range" \lambda), the "fifth force" is negligible, so if you measure the force between masses at large distances (e.g. with satellites and the Earth) it will look just like the normal gravitational force. But at smaller distances, e.g. within a lab, the "fifth force" is not negligible, and in fact the way that it manifests itself is as an extra contribution with a strength \alpha times the gravitational force. You can see this by doing a Taylor expansion in r/\lambda. This explains why, if you measure the gravitational constant in a lab assuming that the only force involved is ordinary Newtonian gravity, you will get a value that differs from larger-scale measurements by a factor of 1 + \alpha.
