Expanding X-ray cavities observed in hot gas atmospheres of many galaxy groups and clusters generate shock waves and turbulence that are primary heating mechanisms required to avoid uninhibited radiatively cooling flows which are not observed. However, we show here that the evolution of buoyant cavities also stimulates radiative cooling of observable masses of low-temperature gas. During their early evolution, radiative cooling occurs in the wakes of buoyant cavities in two locations: in thin radial filaments parallel to the buoyant velocity and more broadly in gas compressed beneath rising cavities. Radiation from these sustained compressions removes entropy from the hot gas. Gas experiencing the largest entropy loss cools first, followed by gas with progressively less entropy loss. Most cooling occurs at late times, $sim 10^8-10^9$ yrs, long after the X-ray cavities have disrupted and are impossible to detect. During these late times, slightly denser low entropy gas sinks slowly toward the centers of the hot atmospheres where it cools intermittently, forming clouds near the cluster center. Single cavities of energy $10^{57}-10^{58}$ ergs in the atmosphere of the NGC 5044 group create $10^8 - 10^9$ $M_{odot}$ of cooled gas, exceeding the mass of extended molecular gas currently observed in that group. The cooled gas clouds we compute share many attributes with molecular clouds recently observed in NGC 5044 with ALMA: self-gravitationally unbound, dust-free, quasi-randomly distributed within a few kpc around the group center.
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