How does the inverted Marcus region explain chemiluminescence?
Marcus theory of electron transfer shows that it is possible for a reaction with a large negative free energy ($-\Delta G$ ) to be slower than one with a less negative free energy. The region where this happens is called the 'inverted' region and is shown in figure (c) in your diagram. You can see that the energy of the products is far below that of the reactants but that there is still a substantial activation energy, that is the energy from the minimum of the R curve to the crossing point of the R-P curve on the left. 

In fig (A) this is called the 'normal' region and the free energy is close to zero and there is still an activation barrier, in (b) the reaction is slightly exothermic has an activation energy and is still in the 'normal' region. What is missing is the case when the reaction is activationless, this lies between figs (b) and (c) and occurs when the P curve crosses the R curve at its minimum. In this situation the reaction has its maximum rate constant. 

The figure shows some data illustrating the rate constnats for electron transfer in the normal(left side) activationless (top) and inverted inverted (right) regions vs $-\Delta G$. The data is for electron transfer from bases in DNA to different intercalated dyes. The fitted Marcus curve is the solid line one from classical Marcus theory the other from a quantum calculation.

Hopefully this will description allow you to follow the paper you have been reading. (You may also want to  look at BBA v811, p265, 1985)

It was not until late 1980's that Marcus theory was confirmed experimentally, first in the solid state and then with chemically linked donor and acceptor molecules as this prevents diffusion, which had hitherto greatly confused interpreting data, and also meant that the inverted region could not be observed.



The energetics for an electron transfer reaction 

$$\ce{D^* +A \rightarrow D^+ + A^-  }$$ is 

$$\Delta G = E^o_{D^+/D} -E^o_{A/A^-}-E_{s_1} +\epsilon_r$$ 

for an excited state $\ce{D^*}$ with energy $E_{s_1}$ and electrode potentials $E^o$. Marcus in the paper also includes a solvation term in a solvent, of dielectric constant $\epsilon$ to account for change in redox. This term can often be neglected but is

$$\epsilon_r = \frac{\Delta e^2}{4\pi \epsilon_o}\left (\left(\frac{1}{r}-\frac{1}{2a}\right)\frac{1}{\epsilon}-\frac{1}{\epsilon_ar}\right)$$

where $\epsilon_a$ is the dielectric constant of acetonitrile, $2a$ is radius of donor + acceptor and r their separation. 

The rate constant is calculated from
$$ k_\mathrm{et} = k^0\exp\left(- \frac{(\Delta G +\lambda)^2}{4\lambda k_\mathrm{B}T}\right)$$

where $\lambda$ is the reorganisation energy and is the vertical (positive) energy between the minimum of the acceptor potential and the donor potential. Its is determine experimentally from the maximum in the rate vs $-\Delta G$ plot as is $k^0$. In the plot $\lambda \approx 1~\mathrm{eV}$ and $k^0 \approx 10^{13} \pu{s^{-1}}$.

The chemiluminescent reaction can be considered as the reverse of the example given.