Earlier quoted context omitted.
If an object is measured to be a billion light years away, that means that light from that object takes a billion years to get here.
Right... but how do you know it takes a billion year to get here?
Measuring distances is an always active part of astronomy, as I understand it, but it starts with parallax.
See this page for more info: https://en.wikipedia.org/wiki/Cosmic_distance_ladder