A new era of observational surveys that are both deep and wide is poised to revolutionise our understanding of galaxy evolution, by enabling, for the first time, statistical studies of the low-surface-brightness (LSB) Universe. While largely inaccessible in past wide-area surveys like the SDSS (due to their lack of depth), the uncharted LSB regime holds the key to a complete understanding of galaxy evolution. While small, deep surveys and new instruments have long hinted at the existence of a rich population of LSB galaxies below the surface-brightness limits of surveys like the SDSS, the mechanisms that create these galaxies remain unexplored. We use, Horizon-AGN, a cosmological hydrodynamical simulation to study how and why low-surface-brightness galaxies (LSBGs;  mu > 23 mag arcsec^-2), and in particular, the recently studied population of ultra-diffuse galaxies, form and evolve over time. For stellar masses greater than 10^7  MSun, LSBGs contribute 85, 10 and 11 per cent of the local number, mass and luminosity densities respectively. When controlled for stellar mass, today's LSBGs have similar dark-matter fractions and angular momenta to their high-surface-brightness (HSB) counterparts but larger (x 2.5) effective radii and lower (< 5% vs 30%) star-forming gas fractions. Interestingly, LSBGs originate from the same progenitors as HSB systems at high redshift (z~3). However, LSBG progenitors form stars more rapidly at early epochs. The higher resultant supernova energy injection flattens their gas-density profiles which, in turn, creates shallow stellar profiles that are more susceptible to tidal processes. After z~1, harassment and tidal heating steadily expand LSBG stellar distributions and quench star formation by heating cold gas, creating the population of diffuse, gas-poor LSB systems seen today. In clusters, ram-pressure stripping provides an additional mechanism that assists in gas removal in LSBG progenitors. The study of LSBGs will be one of most exciting advances in galaxy evolution in the coming years. This study offers insights into the demographics and properties of a population of galaxies that will have a transformational impact on our understanding of galaxy evolution.