The HI gas content is a key ingredient in galaxy evolution, the study of which has been limited to moderate cosmological distances for individual galaxies due to the weakness of the hyperfine HI 21-cm transition. Here we present a new approach that allows us to infer the HI gas mass $M_{rm HI}$ of individual galaxies up to $zapprox 6$, based on a direct measurement of the [CII]-to-HI conversion factor in star-forming galaxies at $zgtrsim 2$ using $gamma$-ray burst afterglows. By compiling recent [CII]-158 $mu$m emission line measurements we quantify the evolution of the HI content in galaxies through cosmic time. We find that the HI mass starts to exceed the stellar mass $M_star$ at $zgtrsim 1$, and increases as a function of redshift. The HI fraction of the total baryonic mass increases from around $20%$ at $z = 0$ to about $60%$ at $zsim 6$. We further uncover a universal relation between the HI gas fraction $M_{rm HI}/M_star$ and the gas-phase metallicity, which seems to hold from $zapprox 6$ to $z=0$. The majority of galaxies at $z>2$ are observed to have HI depletion times, $t_{rm dep,HI} = M_{rm HI}/{rm SFR}$, less than $approx 2$ Gyr, substantially shorter than for $zsim 0$ galaxies. Finally, we use the [CII]-to-HI conversion factor to determine the cosmic mass density of HI in galaxies, $rho_{rm HI}$, at three distinct epochs: $zapprox 0$, $zapprox 2$, and $zsim 4-6$. These measurements are consistent with previous estimates based on 21-cm HI observations in the local Universe and with damped Lyman-$alpha$ absorbers (DLAs) at $zgtrsim 2$, suggesting an overall decrease by a factor of $approx 5$ in $rho_{rm HI}(z)$ from the end of the reionization epoch to the present.
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