Extreme rainfall is expected to increase under climate change carrying potential socio-economic risks. However, the magnitude of increase is uncertain. Over recent decades, extreme storms over the West African Sahel have increased in frequency with increased vertical wind shear shown to be a cause. Drier mid-levels, stronger cold pools, and increased storm organization have also been observed. Global models do not capture the potential effects of lower-to-mid-tropospheric wind shear or cold pools on storm organization since they parametrize convection. Here we use the first convection-permitting simulations of African climate change to understand how changes in thermodynamics and storm dynamics affect future extreme Sahelian rainfall. The model, which simulates warming associated with Regional Climate Pathway 8.5 until the end of the 21st Century, projects a 28% increase of the extreme rain rate of MCSs. The Sahel moisture change on average follows Clausius-Clapeyron scaling, but has regional heterogeneity. Rain rates scale with the product of time-of-storm total column water (TCW) and in-storm vertical velocity. Additionally, pre-storm wind shear and convective available potential energy both modulate in-storm vertical velocity. Although wind shear affects cloud-top temperatures within our model, it has no direct correlation with precipitation rates. In our model, projected future increase in TCW is the primary explanation for increased rain rates. Finally, although colder cold pools are modelled in the future climate, we see no significant change in near-surface winds, highlighting avenues for future research on convection-permitting modelling of storm dynamics.