How do we know galaxy GN-z11 is as far away as it is? The news media recently reported that galaxy GN-z11, formed just 400 million years after the big bang, is much further away than originally thought at perhaps 13.2 billion light years.  They say, 


  The key to the discovery was precisely measuring the shift of the
  galaxy's light into longer, redder wavelengths, which correspond to
  how far the photons had traveled before reaching Hubble's eye.
  (Reuters).


So, suppose I'm standing next to, but safely away from, some train tracks.  As the train approaches, sounding its whistle constantly, the pitch will appear to increase until it reaches me, upon which I will briefly hear the whistle's 'rest pitch' (the pitch of the whistle as if I were aboard the train), then it would decrease as the train moves away.  Thus, the lower the pitch, the further away the train is.  I hope that's right.  So, if we know the rest frequency of galaxy GN-z11, its measured frequency indicates how far away it is from earth. Trouble is, how do we know what the rest frequency of light coming from this distant galaxy is?
So GN z-11 is the latest "furthest away" galaxy. It has a claimed redshift measurement of $z=11.1$, meaning that we are seeing the light it emitted about 400 Myr after the big bang (dependent on an assumed set of cosmological parameters).

To get the redshift measurement, the discoverers used grism (relatively low resolution) spectroscopy in the near infrared. What they were looking for is the rest-frame Lyman alpha continuum break, which would be redshifted into this wavelength range.

The Lyman alpha continuum break is caused by the absorption of nearly all ultraviolet photons with wavelengths smaller than 121 nm. These high energy photons are capable of photoionising the neutral hydrogen present in the intergalactic medium at a range of lower redshifts. This neutral hydrogen is present in abundance at redshifts greater than 6, as it had not yet been re-ionised by quasars and starlight. The consequence is that no light is expected to reach us from rest-frame wavelengths shortwards of 121 nm, but a galaxy's light can reach us from rest-frame wavelengths that are longer than this. When one observes the spectrum, we see flux at long wavelengths which suddenly cuts off at shorter wavelengths. The wavelength of the break is $\lambda = 121 \times (1+z)$ nm, where $z$ is the redshift.

It is this Lyman alpha continuum break that has been identified at an observed wavelength of 1470 nm. This leads to the redshift estimate of $z= (1470/121)-1 = 11.1$.

The details are presented in Oesch et al. (2016). The continuum break is the only thing visible in the spectrum. The authors are confident that this is what it is because their previous broadband photometry had given them an estimated redshift of $&gt;10$ (which is why they observed this candidate in the first place).