The redshift of the CMB is not measured, it is calculated.
The brief explanation is that as the universe expands and cools, it becomes energetically favourable to form bound atoms. The reduction in free electrons allows the universe to become transparent and photons escape as a blackbody radiation field and eventually form the CMB.
The temperature at recombination, $T$, can be calculated using well understood physics (see https://en.m.wikipedia.org/wiki/Recombination_(cosmology)) and if the current temperature $T_0$ is known, then the redshift is simply given by $ z = T/T_0 -1$.
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