How could inflation affect the CMB? This is a naive question and I've just started reading inflation on my own. So the question may not be well-posed.
The energy scale at which inflation took place is much higher than the scale at which photon was decoupled. If that is so, how did the inflationary perturbations be responsible for fluctuations in the CMB?
 A: 
The CMB is the snapshot of the decoupling of photons at 380.000 years , i.e. the photons stop hitting the mess of quark gluon  etc and bouncing around, but escape without interactions from then on, just with the shift in frequency due to the expansion.Mass as we know it in the labs forms after that, originally with hydrogen.
The CMB is extremely uniform, and this uniformity cannot be thermodynamic. Have a look at another answer here. Nevertheless fluctuations appear at the level of 10^-5. The hypothesis in the standard Big Bang model is that after the infaltion period there existed accumulations of energy due to quantum fluctuations, which end up as seeds of the galaxies  we see at our time. These are gravitionally bound energy, otherwise known as changes in space curvature which are embedded in space time, and expanded with the continuous expansion. The photons while decoupling, found more empty space at the flat sites of this curvature, and more of them go through to be registered at the CMB than from the direction of the tight space curvature locations which will evolve into galaxies and clusters of galaxies. 
So the differences in energy density do not play a role, it is that the enegy density fluctuates that is recorded by the CMB photons.
A: There are B-modes predicted. The inflationary cosmos was a de Sitter spacetime with a large cosomological constant, or really parameter $\Lambda(\phi,~\dot\phi)$. A perturbation on this metric can produce gravitational waves. In the inflating spacetime there are quantum fluctuations of the inflaton field, and these produce vacuum gravitational waves. The image below is animated and should show a circular polarized gravitational wave. Gravitational radiation has two degrees of polarization (helicity = 2) and in general it is circular. 

After inflation, which the system falls to the lower energy vacuum, or physical vacuum, radiation and matter is generated. This couples with the gravitational radiation just produced in the $10^{-35}$ to $10^{-32}$ second inflationary period. This induces local anisotropies in the observed distribution of matter. The gravitational radiation is converted into acoustical waves with these circular properties. This patter is then left on the CMB at the surface of last scatter.
In 2015 the BICEP2 released data indicating this. The data was confident to 6-$\sigma$. The measurements are see in the diagram below. However, it was found that dust in the Milky Way galaxy can induce this circular polarization of photons. Analysis caused the data to fall below 3-$\sigma$ confidence. BICEP3 is from my understanding on line and the enhanced ability should separate out inflationary signal from dust.

