They weren't using speed. They were watching how fast a clock ticked back then, the clock being a quasar. The article was light on the details, but the difficulty seemed to be that quasar's aren't reliable clocks, so they needed to develop a statistical model to measure the ticking accurately. Developing the model was actual science being reported on. It happened to agree with Einstein's predictions.
But even if they were using speed, gravity doesn't slow down light. It redshift's it. So there is something you could compare your speed to back then.
Finally, we are comparing how fast time ticks for us now versus how fast a quasar ticked back then. I don't know if the quasar it still around, but if it is it's probably a very different beast now. In any case we are comparing two things that don't exist at the same time. In fact, my guess is it's downright difficult compare yourself to something that exists at the same time, because the speed of light is finite. That means by the time you receive the measurement, whatever you've measured has moved on.
Which universe are they talking about? Most sources I know of cite 13.78 billion years for our universe, and the quoted 1.5 billion years is definitely over 10% of that.
When we look at things far away from us, we're seeing their state in the past. Quasars that are 10 billion light years away took 10 billion years for their light to reach us, so right now we are seeing their images 10 billion years ago.
Quasar is bright light powered by gas spiraling at a high velocity into a black hole. When things revolving around a black hole, the light given out varies. The light of the gas spinning behind the black hole is blocked and the light is shown again when the gas spinning on the side of the black hole facing us. When we're looking at it, it kind of blinks periodically in constant rate. The blinking rate can serve as clock ticks.
The people of the paper figure out a way to compute the blinking rate of the quasars statistically. They look at hundreds of quasars at 12 billion light years away, i.e. looking at their state 12 billion years ago. Compute their blinking rate from twenty years of telescope observation. Compare that to the quasars close to us. They found that the far away quasars blink 5 times slower, i.e. they're blinking 5 times slower 12 billion years ago. The universe is about 13.7 billion years old, so the conclusion is that time ran slower at the early age. It actually fits the prediction from Einstein's general relativity.
Instead of whatever tiny fraction of a second your telescope would enable you to look into the past, the telescopes looking at whole-ass galaxies in the early universe that appear to be "running slower".