We quantitatively investigate the dependence of central galaxy HI mass ($M_{rm HI}$) on the stellar mass ($M_ast$), halo mass ($M_{rm h}$), star formation rate (SFR), and central stellar surface density within 1 kpc ($Sigma_1$), taking advantage of the HI spectra stacking technique using both the Arecibo Fast Legacy ALFA Survey and the Sloan Digital Sky Survey. We find that the shapes of $M_{rm HI}$-$M_{rm h}$ and $M_{rm HI}$-$M_ast$ relations are remarkably similar for both star-forming and quenched galaxies, with massive quenched galaxies having constantly lower HI masses of around 0.6 dex. This similarity strongly suggests that neither halo mass nor stellar mass is the direct cause of quenching, but rather the depletion of HI reservoir. While the HI reservoir for low-mass galaxies of $M_ast<10^{10.5}M_odot$ strongly increases with $M_{rm h}$, more massive galaxies show no significant dependence of $M_{rm HI}$ on $M_{rm h}$, indicating the effect of halo to determine the smooth cold gas accretion. We find that the star formation and quenching of central galaxies are directly regulated by the available HI reservoir, with an average relation of ${rm SFR}propto M_{rm HI}^{2.75}/M_ast^{0.40}$, implying a quasi-steady state of star formation. We further confirm that galaxies are depleted of their HI reservoir once they drop off the star-formation main sequence and there is a very tight and consistent correlation between $M_{rm HI}$ and $Sigma_1$ in this phase, with $M_{rm HI}proptoSigma_1^{-2}$. This result is in consistent with the compaction-triggered quenching scenario, with galaxies going through three evolutionary phases of cold gas accretion, compaction and post-compaction, and quenching.
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