One of the problems is that nobody is saying what kind of coordinate system they are using to define those distances and speeds. Then, to make matters worse, journalists start doing their own calculations using some of these numbers and "calculating" the rest using high school math that doesn't really apply. So you get weird and inconsistent results.
From what I've understood (but this may be completely wrong), you have two different coordinate systems for looking at cosmological events:
You can use a coordinate system (for time AND space) in which the speed of light is constant everywhere (except for local dilations close to masses). In that case, distant objects have not aged as much as we have because they are traveling at high speed relative to us. Distant objects don't just appear younger because light had to travel all the way here, they actually still are younger than us even if you take that travel time into account. At some distance away from us, the big bang has only just happened and time is passing by extremely slowly because those parts are moving away from us at almost the speed of light. But no matter how far, nothing is moving away from us at more than the speed of light. Extremely distant objects have only just begun their big bang acceleration, and will never even reach our age because their speed will approach the speed of light and time will grind to a halt.
Alternatively, you can use "cosmological time" in which the standard for time is local time everywhere. In this model (which is just a choice of time-space coordinates and just as valid as any other), the universe has the same age everywhere BUT the speed of light changes because "space itself is expanding". Actually, light still has the same local speed but space between us and that local bit of space is increasing so light appears to approach us at a slower speed. Also, objects can travel away from us faster than the speed of light (they aren't locally, but they are when measured from here, because space betweeen us and them is expanding). Extremely distant objects are moving away from us so quickly, or space between us is expanding so rapidly, that their light will never get here. Every time light has moved towards us a little bit, the space between us has increased by more than that amount.
To clarify: in the first coordinate system we will never see certain events because they will never happen in our time frame (time grinds to a halt as they approach the speed of light), in the second coordinate system the events have already happened long ago but the light will never reach us. Either way, we have no way of ever observing those events.
The second system, using cosmological time, is the one usually preferred by cosmologists because it makes more sense to them to consider the whole universe at the same age. It doesn't treat our position differently from any other. I personally prefer the first view because it's more intuitive for someone with just a basic grasp of special relativity. But once again, both are really equally valid choices of coordinates (that's all they are).
The GRB you were talking about happened long ago and far away, but still close enough for us to observe it.
Now I'm just trying to make sense of the numbers, and which coordinate system they are using. Maybe it happened at a real distance of 13.1 billion light years away, and it took light 13.1 billion years to get here. That makes sense in the first coordinate system. Or maybe the source is now 13.1 billion light years away, but it was a lot closer when the burst happened, only space has expanded between us making it harder for the light to reach us.
I'm not exactly sure, I hope someone else can tell us what the numbers really mean.
