Star-forming galaxies can be transformed into passive systems through a multitude of mechanisms that quench star formation, such as the halting of cold gas accretion (known as starvation) and the rapid removal of gas in AGN-driven winds. However, it remains unclear which mechanism is the most significant, primary driver of the star-forming–passive bimodality. Leveraging on the statistical power of the Sloan Digital Sky Survey, we have investigated how galaxy quenching depends on both the internal properties of galaxies (i.e. stellar mass) and external factors (i.e. environment), how galaxy quenching has evolved across cosmic time and how quenching operates within galaxies. Building upon an innovative technique for assessing the relative impact of different quenching mechanisms through comparisons of the levels of chemical enrichment in star-forming, green valley and passive galaxies, we have analysed the chemical properties of tens of thousands of galaxies in the local Universe to unveil the primary quenching mechanisms in galaxies. We find that the significant difference in stellar metallicity between passive galaxies and their star-forming progenitors implies that for galaxies at all masses, quenching must have involved an extended phase of starvation. In order to best match the observed properties of local passive galaxies, some form of gas ejection has to be introduced in our models, with outflows becoming increasingly more important with decreasing stellar mass. Through an analysis of the local population of green valley galaxies, we find that quenching operates more slowly in the local Universe than at high redshift. By separating star-forming, green valley and passive galaxies, we further find that the environment leaves a much weaker imprint on the stellar populations of galaxies than was previously thought. Satellite galaxies are only marginally more metal-rich and older than central galaxies of the same stellar mass, with stellar metallicities that show only a weak dependence on halo mass, local overdensity and projected distance from their central. Finally, we find, using integral field spectroscopy from the SDSS-IV MaNGA galaxy survey, that passive galaxies are substantially more metal-rich than star-forming galaxies at all radii, with the stellar metallicity difference decreasing with increasing radial distance. Therefore, starvation is a primary driver of quenching at all radii in galaxies, playing a prominent role in quenching the central regions of galaxies, but playing an increasingly less important role in quenching their outskirts.