The reionization of the intergalactic medium (IGM) is the last major phase transition undergone by the Cosmos, and constitutes one of the few unexplored frontiers in observational cosmology. Although several evidences point to an ionizing photon background at these epochs dominated by star-forming galaxies (SFGs), little is known yet about the exact conditions that allowed Lyman Continuum (LyC) photons to escape from galaxies, as well as about the physical properties of the cosmic reionizers.

Following an observational approach, in this thesis, we have investigated the physics of LyC escape in an unprecedented, statistical way, based on Hubble Space Telescope observations from the Low Redshift Lyman Continuum Survey (LzLCS). Sources with high escape fraction of ionizing photons (fesc) are characterized by high, compact star-formation, a highly ionized interstellar medium (ISM) and prominent Lyman-alpha emission. The mechanisms of LyC leakage were studied by the analysis of down-the-barrel absorption transitions, features probing the cold, low-ionized gas in the ISM. The spectra of Lyman Continuum emitters (LCEs) show systematically weak absorption lines with non-zero residual fluxes, the latter tracing channels in the ISM with low-columm-density throughout LyC radiation escapes (originated after stellar winds or supernovae, presumably).

Using LzLCS data, we found that strong LCEs show low values of dust-attenuation and consequently steep, negative (blue) slopes in the ultraviolet continuum (beta-slope). In addition, we report a tight connection between beta-slope and the product of the escape fraction and the ionizing production efficiency. Integrating this relation over the number density of galaxies at the EoR shows that SFGs provided sufficient LyC photons to exceed recombinations at z = 7 - 8.

Finally, by invoking radiative transfer arguments, we provide empirical recipes to predict fesc from simple measurements of the absorption lines and dust-attenuation in galaxies. We apply this method for the first time to a sample of SFGs at 3 < z < 5 with deep, rest-ultraviolet spectra from the VANDELS ESO public spectroscopic survey. By extrapolating our conclusions to the EoR, our results suggests that the ionizing output at the EoR was most probably dominated by the low-mass and dustless galaxies which, according to recent observations from the James Webb Space Telescope, seem to be common properties in the high-z Universe.