This is the history of the universe in the Big bang model.
It shows the currently accepted model . The CMB appears at 380.000 years after the origin of the Big Bang. Before that , up to
10^-32seconda from the origin of the Big bang, our theoretical knowledge of particle physics is heavily used. The CMB happens when due to inflation the universe cools enough so that photons can decouple from the rest of the particle soup and thus keep a snapshot of the distribution of matter at the time of decoupling.
The detailed, all-sky picture of the infant universe created from nine years of WMAP data. The image reveals 13.77 billion year old temperature fluctuations (shown as color differences) that correspond to the seeds that grew to become the galaxies. The signal from the our Galaxy was subtracted using the multi-frequency data. This image shows a temperature range of ± 200 microKelvin
In has small non uniformity from the black body radiation curve
the root mean square variations are only 18 µK
on a 2.7K black body radiation curve, and those are the features in the image above.
It carries information on the seeds of galaxies and cluster of galaxies. In this sense it validates the Big Bang model.
So it validates the Big Bang model, together with a number of other observations.
you ask in the title:
Age of the CMB: How do we know?
The CMB is heavily red shifted because of the expanding universe, and one way of finding the age of the universe is in using the standard model of particle physics and thermodynamics to get at the Black body radiation curve at the time of decoupling. There exist a specific model which cosmology uses to define the chronology of the Big Bang, and everything is consistent within the parameters of this model and the time lines quoted depend on it.
The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parametrization of the Big Bang cosmological model in which the universe contains a cosmological constant, denoted by Lambda (Greek Λ), associated with dark energy, and cold dark matter (abbreviated CDM). It is frequently referred to as the standard model of Big Bang cosmology because it is the simplest model that provides a reasonably good account of the following properties of the cosmos:
the existence and structure of the cosmic microwave background
the large-scale structure in the distribution of galaxies
the abundances of hydrogen (including deuterium), helium, and lithium
the accelerating expansion of the universe observed in the light from distant galaxies and supernovae
So the chronology of the CMB comes out from the "fit" to the observations.