It has a so-called "redshift" of 6.29, which translates to a distance of about 13 billion light years from Earth. ... "This burst smashes the old distance record by 500 million light years," said Dr Daniel Reichart of the University of North Carolina, who has been leading the measurement of its distance.
How do we know it's the farthest? Determining the distance to astronomical objects was one of the most important discoveries of the 20th century. Edwin Hubble (after whom the Hubble Space Telescope is named) discovered that galaxies that are farther away from us (based upon earlier measurements of distance) are also moving away from us faster (greater speed). The speed is directly measured by the Doppler Effect / Shift - the same thing that makes the siren of an abulance driving past change in pitch. In GRBs and other things in the sky, we measure how the color of the light changes and use this to determine the velocity, which cosmologists translate into redshift. The velocity to redshift conversion is exact; the redshift or velocity to distance conversion is still under debate. The current best guess is 71±4 (km/s)/Mpc.
The Swift telescope used in this study can actually detect GRBs even farther away than that, if they happen. Light takes a while to get here, so when you look at objects far away, you're looking at long ago in the past. Think about sending letters by snail mail. If you send one in the same town, it might get there the same day, so if you get a letter from the same town, the person probably wrote it about the day before. If you get a letter from a few states away, the letter's a few days old. A letter from the other side of the country might be a week old, and any news it has will be that long out of date. Even farther away, a letter from Europe would take a couple weeks.
Swift can see very dim objects, dim = far away = long ago = soon after the Big Bang (200 million years after, they guess). We don't yet know how soon after the Big Bang stars started to form. If there were no stars, then none of them would've been dying and forming GRBs! Can you imagine a universe that hasn't yet formed stars?
1 Regarding Wikipedia, caveat emptor (let the buyer beware). While the vast majority of their information is correct and well-written, there are occasional errors that slip through. This is the result of it being open-source: anyone can contribute, including YOU, and only other editors can stop mistakes and vandalism.