Inflation is modeled by the dynamics of a scalar field.  One then introduces perturbations in the scalar field.  One then quantizes the perturbations in terms of quantum fields.  One can then calculate the two-point function of these field variables.  And one can then extract the “Power spectrum” from this 2-pt function.  And…. eventually this produces a description of the CMB that can be compared, very favorably, with observations.