We present a study of ~100 high redshift (z~2-4) extremely strong damped Lyman-alpha systems (ESDLA, with N(HI)>0.5x10^22 cm^-2) detected in quasar spectra from the Baryon Oscillation Spectroscopic Survey SDSS-III DR11. We study the neutral hydrogen, metal, and dust content of this elusive population of absorbers and confirm our previous finding that the high column density end of the N(HI) frequency distribution has a relatively shallow slope with power-law index -3.6, similar to what is seen from 21-cm maps in nearby galaxies. The stacked absorption spectrum indicates a typical metallicity ~1/20th solar, similar to the mean metallicity of the overall DLA population. The relatively small velocity extent of the low-ionisation lines suggests that ESDLAs do not arise from large-scale flows of neutral gas. The high column densities involved are in turn more similar to what is seen in DLAs associated with gamma-ray burst afterglows (GRB-DLAs), which are known to occur close to star forming regions. This indicates that ESDLAs arise from lines of sight passing at very small impact parameters from the host galaxy, as observed in nearby galaxies. This is also supported by simple theoretical considerations and recent high-z hydrodynamical simulations. We strongly substantiate this picture by the first statistical detection of Lya emission with <L>~(0.6+/-0.2)x10^42 erg/s in the core of ESDLAs (corresponding to about 0.1 L* at z~2-3), obtained through stacking the fibre spectra (of radius 1 corresponding to ~8 kpc at z~2.5). [truncated]
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