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 & the equations.