They use the Lamda-CDM model which outputs the rate of expansion of the universe at every moment in past present and future. You measure the amount of light+matter+dark matter+dark energy that your universe has, plug those values into the Friedmann equation, and it spits out the rate.
You can try out an online calculator yourself! It already has those values filled in, all you need to do is enter the z value - the “redshift” - and click generate. So for example when you hear in the news something like “astronomers took a photo of a galaxy at redshift 3”, you put in 3 for “z”, and you see that the galaxy is 21.1 Gly (billions light years) away! That’s the “comoving distance”, a convenient way to define distance on cosmic scales that is independent of expansion rate or speed of light. It’s the same definition of distance that gives you that “46 Gly” value for the size of observable universe. But the light from that galaxy only took 11.5 Gyr to reach us. The universe was 2.2 Gyr old when the light started. So the light itself only traveled 11.5 Gly distance, but that distance is 21.1 Gly long right now because it kept expanding behind the photon.