This is to agree with ProfRob and add a few more details.
Black body radiation, also called cavity radiation, has the special feature that many of its features depend on only one thing: the temperature. For example:
- energy per unit volume $u = 3 a T^4$
- entropy per unit volume $s =
4 a T^3$
- pressure $p = a T^4$
- power per unit area incident on a
surface $I = \frac{1}{4}uc
= \sigma T^4$
where
$$
a = \frac{4 \sigma}{3 c}
$$
and $\sigma$ is the Stefan Boltzmann constant.
This means that if this kind of radiation is falling on a detector, then the energy flux, also called intensity, is not a freely variable parameter: once the temperature is given, so is the intensity $I$.
Suppose the detector is not perfectly efficient. In this case the signal strength will depend on the efficiency. One can model this by supposing there is an absorbing layer between the incident radiation and a perfect detector. In this case one of two things can happen. If the absorbing layer is itself passive then eventually it will reach the temperature of the radiation and then it has no net effect so the signal amplitude goes to full strength. Or, if the absorbing layer carries some energy away (e.g. by producing an electric current) then the detector gets a weaker signal.
This inefficiency issue can be studied beforehand for any given detector, and thus the detector is calibrated for amplitude. Once calibrated, it can give a precise reading for incident amplitude with the known inefficiency accounted for.
It is a non-trivial fact about General Relativity that, in an expanding space, all these properties of thermal radiation are preserved, and thus, amazing as it may seem, the amplitude of the CMB is not a function of distance from the last scattering surface except through the way cosmic expansion affects the temperature and all the other properties together.
Added note
This added note is to address the fact that one can have a body such as a star or an electric filament light bulb that emits radiation with a black body spectrum, and that radiation diminishes in intensity with distance from the source. Such radiation can legitimately be called 'thermal' but it should not be called 'cavity radiation' and it is a moot point whether or not the terminology should allow it to be called 'black body radiation' because it is not homogeneous, so it does not have all the properties of cavity radiation described above. The CMB is (to very good approximation) cavity radiation and at each spatial location there exists a local reference frame in which it is isotropic.