Introduced species threaten native biodiversity, but whether exotic species can competitively displace native species remains contested. Building on theory that predicts multi-species coexistence based on a competition-colonisation tradeoff, we derive a mechanistic basis by which human-mediated species invasions could cause extinctions through competitive displacement. In contrast to past invasions, humans principally introduce modern invaders, repeatedly and in large quantities, and in ways that can facilitate release from enemies and competitors. Associated increases in exotic species' propagule rain, survival and competitive ability could enable some introduced species to overcome the tradeoffs that constrain all other species. Using evidence from metacommunity models, we show how species introductions could disrupt species coexistence, generating extinction debts, especially when combined with other forms of anthropogenic environmental change. Even though competing species have typically coexisted following past biogeographic migrations, the multiplicity and interactive impacts of today's threats could change some exotic species into agents of extinction.