First of all I would call it an inference or estimate rather than a measurement, and even then it is not an easy inference.
We can work the distance to ‘nearby’ stars by noticing that they shift their apparent position in the sky ever so slightly as we orbit the Sun. By measuring these shifts in position and knowing the diameter of the Earth’s orbit around the Sun, we can work out the distance to these ‘nearby’ stars.
We notice that some stars have variable brightness. Over a few weeks their brightness changes and importantly we can relate their absolute brightness to the period of variable brightness.
Now the argument goes – if we see a star of this variable type in a very distant region, then if we measure the period of its brightness oscillation, then we know how bright it is absolutely. We then measure how bright it appears – and the ratio tells us how far away it is.
Using this technique we can estimate the distance to stars that are a very long distance away.
Now we notice that the spectrum of these distant stars is shifted as if the stars were moving away from us. In fact every distant star appears to be moving away from us!
The speed of retreat appears to be proportional to their distance from us.
We extrapolate back to find that around 13.7 billion years ago, we would all have been in the same place – the big bang.
And that is how you estimate the age of the Universe using light. It is not easy or straightforward, but the inferences have been checked and double checked – and they do seem robust.
Image: The anisotropies of the Cosmic microwave background (CMB) as observed by Planck. The CMB is a snapshot of the oldest light in our Universe, imprinted on the sky when the Universe was just 380 000 years old. It shows tiny temperature fluctuations that correspond to regions of slightly different densities, representing the seeds of all future structure: the stars and galaxies of today.