Image source: [1]
In the first place, since we know quite accurately the life cycle of the stars we can put a lower dimension to the age of the universe from the oldest objects. This is a quick way to see if a universe model is "good" or not according to the age of the universe it predicts.
Let us now go to the calculation itself. As we look at the cosmos as we look further we also look to the past because of the time the light needs to reach us. Also due to the expansion of the universe the light is stretching resulting in the known redshift. The oldest light is the cosmic microwave background originating from the same Big Bang. The idea is that we can relate the redshift / expansion of the universe to the elapsed time. For this we must elaborate a model estimating the density of matter, radiation and dark energy that reproduces the observations of the expansion of the universe. Once we have this adjusted, the model tells us how much time must have elapsed to arrive at how we are today with the expansion that there is today or, if you prefer the more poetic version, reverse the passage of time in the model until the Big Bang And see how long ago it happened.
At present the calculation puts the age of the universe around 13700 million years. The oldest light, the WBC, was "liberated" when the universe was about 380,000 years old and before it was not free to propagate and the universe was opaque, but as you can see this is a very small amount compared to the total age and In addition knowing it can be incorporated into the calculations.
More details here the link: [2]
