As self-gravitating systems, dense star clusters exhibit a natural diffusion of energy from their innermost to outermost regions, which leads to a slow and steady contraction of the core until it ultimately collapses under gravity. However, in spite of the natural tendency toward so-called core collapse, the globular clusters (GCs) in the Milky Way exhibit a well-observed bimodal distribution in core radii separating the core-collapsed and non-core-collapsed clusters. This suggests an internal energy source is at work, delaying the onset of core collapse in many clusters. Primordial binary stars have been thought for a long time to provide this energy source, but recent analyses have cast doubt upon the corresponding binary-burning mechanism as a viable explanation. Over the past decade, a large amount of both observational and theoretical work has suggested that many stellar-mass black holes (BHs) are retained in typical clusters today and that they play a dynamically-significant role in these clusters throughout their entire lifetimes. Here we review our latest understanding of the formation and evolution of BH populations in GCs and demonstrate that, through their dynamical interaction with their host cluster, BHs can naturally explain the distinction between core-collapsed and non-core-collapsed clusters through a process we call black hole burning.