It's... complicated. I'm not sure I fully understand myself so trying to explain it to another person is probably beyond me. But I'll see what I can tell you.

First, light doesn't change in speed, it changes in wavelength when emitted from something moving relative to you (away from you or towards you). It's kind of like the doppler effect where a sound moving towards you will have one pitch, but when it passes you and starts moving away, the pitch changes. Light is kind of the same.

So light emitted from a source moving towards you turns blue and light emitted from a source moving away from you turns red. The faster the thing is moving towards you or away from you, the more the color of the light changes in that direction.

When we look out much beyond our galaxy, everything we see is red, so everything is moving away from us. We can see the things that are furthest away are redder than the things that are near so the furthest stuff is moving away faster than the closest stuff. And by measuring this "red shift" in color, we can calculate how fast these things are moving away. And by reversing that speed, we can calculate how long it had been since those things were where we are now.

Now I'm not sure if this is exactly how it's done, but logically if you calculate how long it had been since literally everything was all in one spot, that can tell you the age of the universe.

Does that make sense? Kind of like one of those old words problems. A train is moving away from you at 60 mph. It is 45 miles away from you now. How long has it been since the train was at your location?

We’ve observed distant galaxies moving farther away from us. If we calculate the speed they’re going away from us then we can approximate the beginning of the universe. What happened before that time is unknowable.