Monsoon droughts over the Indian subcontinent emanate from failures in the seasonal (June--September) monsoon rains. While prolonged dry-spells ("monsoon-breaks") pervade on sub-seasonal/intra-seasonal time-scales, the underlying causes for these long-lasting anomalies remain elusive. Based on analyses of a suite of observed data sets, we report an ocean-atmosphere dynamical coupling on intra-seasonal time-scales, in the tropical Indian Ocean, which is pivotal in forcing extended monsoon-breaks and causing droughts over the subcontinent. This coupling involves a feedback between the monsoonal flow and thermocline depth in the Equatorial Eastern Indian Ocean (EEIO), in which an anomaly of the summer monsoon circulation induces downwelling and maintains a higher-than-normal heat-content. The near-equatorial anomalies induce strong and sustained suppression of monsoon rainfall over the subcontinent. It is concluded that the intra-seasonal evolution of the ocean-monsoon coupled system is a vital key to unlocking the dynamics of monsoon droughts.