The role of the constant false alarm rate circuitry is to determine the power threshold above which any return can be considered to probably originate from a target as opposed to one of the spurious sources.
Conversely, fewer targets will be detected if the threshold is too high, but the number of false alarms will also be low.
However, in most fielded systems, unwanted clutter and interference sources mean that the noise level changes both spatially and temporally.
More sophisticated CFAR algorithms can adaptively select a threshold level by taking a rigorous account of the statistics of the background in which targets are to be detected.
This is particularly common in maritime surveillance (radar) applications, where the background of sea clutter is particularly spikey and not well approximated by additive white Gaussian noise.