I may be off, but I'd auggest looking up an "atomic clock" and either reading about it or watching a short video explanation.
In short, let's say you built a "clock" that measured time based on how long it took a photon (light particle) to bounce between two sensors (1 bounce = 1 "tick" of the clock). If you began to move the clock through space, the faster you'd move the clock, the longer it'd take the photon to complete one tick because the photon is now moving angularly with the clock. You can take this to the most extreme by theoretically moving the clock at a speed arbitrarily close to the speed of light, where it'd almost never complete the "tick."
I believe that is the principle at play here when they're measuring light from ancient phenomenon like quasars. What they're finding interesting is that, based on their math, effectively the shape of the light wave from those quasars is different than if those quasars were currently happening, which they can use as evidence that these events unfolded at a different "speed" than our current universal conditions.
That's just my best attempt to connect concepts I've studied to what is clearly a far more complex analysis. Still, I hope some of that helps introduce the relevant concepts to interested minds.