We use the eagle simulations to study the connection between the quenching timescale, $tau_{rm Q}$, and the physical mechanisms that transform star-forming galaxies into passive galaxies. By quantifying $tau_{rm Q}$ in two complementary ways - as the time over which (i) galaxies traverse the green valley on the colour-mass diagram, or (ii) leave the main sequence of star formation and subsequently arrive on the passive cloud in specific star formation rate (SSFR)-mass space - we find that the $tau_{rm Q}$ distribution of high-mass centrals, low-mass centrals and satellites are divergent. In the low stellar mass regime where $M_{star}<10^{9.6}M_{odot}$, centrals exhibit systematically longer quenching timescales than satellites ($approx 4$~Gyr compared to $approx 2$~Gyr). Satellites with low stellar mass relative to their halo mass cause this disparity, with ram pressure stripping quenching these galaxies rapidly. Low mass centrals are quenched as a result of stellar feedback, associated with long $tau_{rm Q}gtrsim 3$~Gyr. At intermediate stellar masses where $10^{9.7},rm M_{odot}<M_{star}<10^{10.3},rm M_{odot}$, $tau_{rm Q}$ are the longest for both centrals and satellites, particularly for galaxies with higher gas fractions. At $M_{star}gtrsim 10^{10.3},rm M_{odot}$, galaxy merger counts and black hole activity increase steeply for all galaxies. Quenching timescales for centrals and satellites decrease with stellar mass in this regime to $tau_{rm Q}lesssim2$~Gyr. In anticipation of new intermediate redshift observational galaxy surveys, we analyse the passive and star-forming fractions of galaxies across redshift, and find that the $tau_{rm Q}$ peak at intermediate stellar masses is responsible for a peak (inflection point) in the fraction of green valley central (satellite) galaxies at $zapprox 0.5-0.7$.
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