So if we can see something 15 billion light years away, we actually see it as it looked 15 billion years ago
I'll use a sci-fi example to illustrate this. I think it was the Berserker books in which this happened....
A battle in space would take place. It was over quickly, and the human ships' distress call carried little information about the attack. The rescuing vessels arrived at the scene of the battle to find a field of debris. Well, what to do? They looked at the debris to find out how long ago the battle took place. Since they have faster than light space travel, they then jumped/warped/whatever far enough away from the battle so that the light of the battle hadn't gotten that far yet. They then point very powerful telescopes at the scene of the battle, and they can watch the whole thing 'as it happens' (IE., as the light from the battle starts arriving at their position). If they miss a detail, they just warp a little further out and rewatch the whole thing as the light beams advance ever further out.
The thing to remember is that, while it appears so to our limited perception, light is NOT instantaneous. It takes a certain amount of time for the 'light' from a bulb to reach a wall. (Not much time, to be sure, but when you are crossing galaxies instead of hallways, it becomes VERY significant).
If we look up at the sky and see light that is 15 billion light years away - that light has been coming from its source, spreading ever outward - and finally reaching earth for FIFTEEN BILLION YEARS.