Understanding the climate dynamics at the inner edge of the habitable zone is crucial for predicting the habitability of rocky exoplanets. Previous studies using global climate models (GCMs) have indicated that planets receiving high stellar flux can exhibit climate bifurcations, leading to bistability between a cold (temperate) and a hot (runaway) climate. However, the mechanism causing this bistability has not been fully explained, in part due to the difficulty associated with inferring mechanisms from small numbers of expensive numerical simulations in GCMs. In this study, we employ a two-column (dayside and nightside), two-layer climate model to investigate the physical mechanisms driving this bistability. Through mechanism-denial experiments, we demonstrate that the runaway greenhouse effect, coupled with a cloud feedback on either the dayside or nightside, leads to climate bistability. We also map out the parameters that control the location of the bifurcations and size of the bistability. This work identifies which mechanisms and GCM parameters control the stellar flux at which rocky planets are likely to retain a hot, thick atmosphere if they experience a hot start. This is critical for the prioritization of targets and interpretation of observations by the James Webb Space Telescope. Furthermore, our modeling framework can be extended to planets with different condensable species and cloud types.