Why is it that "With higher CO2, the dry gets drier and the wet gets wetter"?
To my best knowledge, the paper by Held &amp; Soden (2006) is the first paper that discussed the concept of the rich get richer. They used the climate change experiments generated for the Fourth Assessment of the intergovernmental panel on Climate Change or in other words, climate projections using climate models forced with different CO2 scenarios.
One of their findings is that:

Assuming that the lower-tropospheric relative humidity is unchanged
and that the flow is un-changed, the poleward vapor transport and the
pattern of evaporation minus precipitation (E - P) increases
proportionally to the lower-tropospheric vapor, and in this sense wet
regions get wetter and dry regions drier.

To explain, assuming warming earth leads to more evaporation and an increase in the air capacity to carry water vapor, therefore, relative humidity can be assumed unchanged in the lower-tropospheric. Simoustanly, with warmer mid and higher latitudes the act of the atmosphere (through General Circulation) to redistribute the heat is reduced. Thus, more water vapor remains in the regions of evaporation leading to wet regions get wetter and dry regions drier.
Yes, this result is widely accepted in the community.