Simple Summary: The expansion of mutant cells that outgrow and displace normal blood cells is called clonal hematopoiesis (CH). CH starts with the acquisition of mutations in blood stem cells that change their normal behavior and improve their fitness. Low numbers of mutant blood cells are present in many middle-aged and elderly people without noticeable effects. However, CH can progress to leukemia and contribute to diseases in other tissues, including the heart, liver, and pancreas. As these diseases progress and affect health when the number of mutant blood cells increases, preventing and/or reducing the expansion of mutant cells might delay this process. Identifying the factors driving mutant blood cell expansion is thus critical and has garnered heightened interest in the subject. Here, we review and discuss recent progress in the field that suggests that mutant blood stem cells are more resilient than normal cells and thrive in inflammatory conditions. Clonal hematopoiesis (CH), the relative expansion of mutant clones, is derived from hematopoietic stem cells (HSCs) with acquired somatic or cytogenetic alterations that improve cellular fitness. Individuals with CH have a higher risk for hematological and non-hematological diseases, such as cardiovascular disease, and have an overall higher mortality rate. Originally thought to be restricted to a small fraction of elderly people, recent advances in single-cell sequencing and bioinformatics have revealed that CH with multiple expanded mutant clones is universal in the elderly population. Just a few years ago, phylogenetic reconstruction across the human lifespan and novel sensitive sequencing techniques showed that CH can start earlier in life, decades before it was thought possible. These studies also suggest that environmental factors acting through aberrant inflammation might be a common theme promoting clonal expansion and disease progression. However, numerous aspects of this phenomenon remain to be elucidated and the precise mechanisms, context-specific drivers, and pathways of clonal expansion remain to be established. Here, we review our current understanding of the cellular mechanisms driving CH and specifically focus on how pro-inflammatory factors affect normal and mutant HSC fates to promote clonal selection.