probably several ways.
1) redshift. This gives distance, and the speed of light therefore gives time.
2) they know the spectrum of the light in order to claim the stars don't have metals in them. This means they must be first generation stars.
3) other methods I've not thought of yet.
The main idea is indeed "redshift", but at these extreme distances it's not quite the same thing as measuring where the spectral lines have moved to.
These galaxies are far, far too faint to register on a high-resolution spectrometer---a spectrometer has to spread light out, which means you're taking the (few thousand!) photons you collect from the galaxy and schmearing them across 1000 pixels of your CCD. You simply don't see anything in a spectrometer.
What you can do is collect all of your photons on a *few* CCD pixels (which does give a reasonable image, visible above the noise) and then swap a few filters in front of the CCD in order to get some very rough spectral information. It's a spectrum, but too coarse to see spectral lines.
There is one feature that all star-forming galaxies have, though, which is not a line but an "edge". These galaxies emit scads of hard-UV light from young stars, then reabsorb anything that happens to be below the Hydrogen lyman-alpha wavelength of 121.6 nm. So you expect all young galaxies to have this "step" function in their rest frame spectrum, with not much emission below 121.6nm then suddenly lots above. When you're looking at a high-redshift galaxy, say at z=6 or z=7, then this "Lyman break" happens not in the UV but in the IR. Where does it happen? That depends on the redshift.
So: most identification of ultra-high-redshift galaxies is done by looking for galaxies with a "lyman dropout" somewhere in the IR. You look for something that is visible through several longer-IR-wavelength-passing filters, but "suddenly" invisible through a short-wavelength-passing filter.
Now you know at what wavelength the "dropout" appears, and that tells you how far the 121.6nm cutoff has been redshifted, and that tells you the redshift. Technically it's the redshift of the Ly-Alpha absorber, not the UV emitter behind it, but it's fair to assume that they're right next to one another.
You have to be careful---it's possible for a nearby object to happen to have a spectral "step" (having nothing to do with Ly-Alpha) at 1500nm---so there are various followup checks, but that's not my field so I don't claim to know how they all work.
