Understanding the future of extreme temperatures is a critical goal for science and society. Simulations with coupled climate models suggest that hot days over tropical land will warm substantially more than the average day. For example, averaged across models, warming of the hottest 5% of days is projected to be approximately 20% larger than the time-mean warming. Amplified warming of extreme temperatures implies severe impacts on humans and ecosystems, yet the physical mechanisms underpinning this emergent behaviour of numerical models remain unclear. Here, I interpret the response of extreme temperatures over tropical land to climate change using a theory based on convective coupling and the weak temperature gradient approximation. According to the theory, warming is amplified for hot land days because those days are dry: this is termed the “drier get hotter” mechanism. Changes in near-surface relative humidity further increase tropical land warming, with decreases in land relative humidity particularly important. The theory advances understanding of extreme weather in the tropics, and highlights climatological land-surface dryness as a key factor determining how hot days will respond to climate change. Ideas for generalising the theory beyond the tropics will be discussed.