We present an analysis of the levels and evolution of star formation activity in a representative sample of 30 massive galaxy clusters at 0.15 < z < 0.30 from the Local Cluster Substructure Survey, combining wide-field Spitzer/MIPS 24 mum data with extensive spectroscopy of cluster members. The specific SFRs of massive (M > ~ 10 super(10) M sub(middot in circle)) star-forming cluster galaxies within r sub(200) are found to be systematically ~28% lower than their counterparts in the field at fixed stellar mass and redshift, a difference significant at the 8.7sigma level. This is the unambiguous signature of star formation in most (and possibly all) massive star-forming galaxies being slowly quenched upon accretion into massive clusters, their star formation rates (SFRs) declining exponentially on quenching timescales in the range 0.7-2.0 Gyr. We measure the mid-infrared Butcher-Oemler effect over the redshift range 0.0-0.4, finding rapid evolution in the fraction (functionof sub(SF)) of massive (M sub(K) < - 23.1) cluster galaxies within r sub(200) with SFRs > 3 M sub(middot in circle) yr super(-1), of the form functionof sub(SF) alpha (1 + z) super(7.6 + or - 1.1). We dissect the origins of the Butcher-Oemler effect, revealing it to be due to the combination of a ~3 x decline in the mean specific SFRs of star-forming cluster galaxies since z ~ 0.3 with a ~1.5 x decrease in number density. Two-thirds of this reduction in the specific SFRs of star-forming cluster galaxies is due to the steady cosmic decline in the specific SFRs among those field galaxies accreted into the clusters. The remaining one-third reflects an accelerated decline in the star formation activity of galaxies within clusters. The slow quenching of star formation in cluster galaxies is consistent with a gradual shut down of star formation in infalling spiral galaxies as they interact with the intracluster medium via ram-pressure stripping or starvation mechanisms. The observed sharp decline in star formation activity among cluster galaxies since z ~ 0.4 likely reflects the increased susceptibility of low-redshift spiral galaxies to gas removal mechanisms as their gas surface densities decrease with time. We find no evidence for the build-up of cluster S0 bulges via major nuclear starburst episodes.