They are discussing a photon being captured by an electron, changing its energy level, saying that the photon, traveling at c, slows down to 1/137 of c when it encounters the electron. The stuff with the electron changing energy levels is not relevant, because the capture changes the net speed of the electron, which changes its orbital velocity.
In the RS, the IRR for an electron is 128(1+1/9) = 142.
The range of photons between infrared and ultraviolet are
spatially-displaced, just like the electron, so there should not be a region crossing upon their interaction. Space is adding to space, giving more space. (Consider it must be this situation, as if the photon were time-displaced, the relation of space to time constitutes motion, and the photon would pass through the electron, never being captured.)
You can still get the basic idea from the IRR formula. The 128 is the total degrees of freedom. The DOF determines how much a speed is "rotationally distributed" (as Larson puts it), so if you were to measure it, you only see 1/DOF of the original speed, no matter where you measure.
So the electron has already slowed to 1/128c because it has been captured by an atom. A photon has 9 DOF but does not slow because it is being carried by the progression at the speed of light. When the photon is captured by the electron, there is a measured drop in speed of that of the electron DOF + the photon DOF, 128 + 9 = 137. That is probably where the number is coming from.
There is one curiosity about this. Since both are space-displaced motions and time remains constant, the speed measured should be 1/137 FASTER than the speed of light, not slower (more space in the same amount of time). I bet that they are measuring speed as a displacement from the speed of light to get the 1/137, but have not considered the direction of that shift, assuming it must be slower to match conventional theory.