CR inhibits mTOR (mTOR1, I think), and, likely through mTOR, increases the production of mitochondria. Increasing the number of mitochondria in a cell, all else being equal, reduces the voltage across each individual mitochondrial membrane, and thus reduces the amount of reactive oxygen species predicted, and thus the rate of mitochondrial DNA damage. Mitochondrial DNA damage reduces ATP production, reducing the ability of the cell to do its work, and also to produce protein for its own upkeep; and may also lead to apoptosis.

This is a brief summary, but there's lots of evidence IMHO that either mitochondrial proliferation, or else the general downregulation of anabolism, is responsible for the effects of CR.