How does an Echelle spectrograph put such a long, high resolution spectrum into a square-ish format?
Echelle spectrometers operate in the higher spectral orders.  (CRIES+ uses orders 10-60 for IR light, I think.  Not sure about ESPRESSO, but you can count them...)  

These high orders are highly overlapping, so, for example, if the input light is dispersed by the Echelle grating horizontally into a spectrum, then the light at a given point along the spectrum is has components contributed from several different orders.  But, since each contribution is a different wavelength from each order, the orders can be spread out vertically using a second grating, the "cross disperser", conveniently stacking them up for the camera.  You can see these components in this diagram of the ESPRESSO spectrograph.

TL;DR:  Each horizontal line you see is a portion of a single spectral order, and the orders are spread vertically by a cross disperser grating, oriented roughly perpendicularly to the first (Echelle) grating.

I found this presentation with more details &amp; the equations.