windy miller said:
"Modes of a given (comoving) wavelength should “freeze out” upon first crossing the Hubble radius during inflation, remain (nearly) constant in amplitude while longer than the Hubble radius, and then resume oscillation upon reentering the Hubble radius. The tem- poral oscillations of modes with nearby wavelengths are therefore coherent [10], giving rise to a sharp pattern of peaks and troughs in the cosmic microwave background (CMB) power spectrum.
First, this is really an "A" level topic, not an "I" level topic. This is to be expected with most aspects of inflationary cosmology. I'll try to give a brief "I" level answer here, and then give a reference (which is most definitely an "A" level reference) that goes into more detail. I emphasize that this description is heuristic and will leave a lot of loose ends; that's unavoidable.
The "modes" being referred to are quantum fluctuations in the spacetime curvature of the universe during inflation. Classically, the curvature is constant on any spacelike surface of constant FRW coordinate time; but once we bring in QM we have to pay attention to the uncertainty principle, which in this context says that the spacetime curvature can't be exactly a single value everywhere. There will be unavoidable fluctuations. These fluctuations will occur with widely varying wavelengths, and as inflation proceeds those wavelengths will be "stretched" by enormous amounts because of the rapidly accelerating expansion of the universe.
Once a fluctuation's wavelength is stretched so much that it becomes longer than the Hubble radius, one end of the wave can no longer causally communicate with the other end. That means the amplitude of the wave--its deviation from the "average" value of spacetime curvature--becomes "frozen": it stops fluctuating (which it had been doing up until that point--that is what "quantum fluctuations of spacetime curvature" means). Explaining in more detail
why this happens is beyond the scope of an "I" level discussion, and also beyond my expertise; all I can do is describe
what happens in the phenomenon referred to.
At some point after inflation ends, the Hubble radius will have increased to the point where it becomes larger than the wavelength of the "frozen" wave. At that point, the two ends of the wave can causally communicate with each other again. However, conditions in the universe are now very different: instead of all the energy density (which is what drives the average spacetime curvature) being contained in the "false vacuum" state of the inflaton field, it is now contained in the hot, dense, rapidly expanding matter and radiation that fills the universe. That drastically changes the dynamics of the wave: its amplitude, while it will not remain exactly fixed (briefly, overdense regions will become more overdense, so waves with amplitudes over the average curvature will have their amplitudes increase; underdense regions, waves with amplitudes under the average curvature, will do the opposite), will fluctuate, if at all, by only a very small amount. So the overall
pattern of fluctuations over the universe will be basically the same as the pattern at the end of inflation (the time evolution just described is easy to run in reverse, so we can evolve the pattern we observe today back to the pattern at CMB formation, and from there the amplitudes stay constant back to the end of inflation), which means it carries information about the quantum fluctuations that took place during inflation (and therefore about the inflation process itself).
At some point further on, the CMB is formed, and the pattern of spacetime curvature fluctuations described above leaves an "imprint" of a similar pattern of fluctuations in the CMB radiation. Which we, in turn, observe billions of years later with the Planck satellite.
The following reference (reference 10 in the Guth et al. paper linked to in the OP) is a better one for understanding the specific phenomenon described above:
https://arxiv.org/abs/hep-ph/0309057
See in particular Fig. 2 and the discussion surrounding it.