The actual study says that women with certain genes known to increase the risk of breast cancer tend to have larger breasts—which is a slightly different statement. They took information from using 16,165 contributors to 23AndMe, used the self-reported information (bra sizes, age, cancer history, pregnancy history, currently nursing or not, etc), and found the correlations mentioned in their conclusions. They used cup size, which they admit is problematic. And they mention a study that supports CBS' headline a little better: "Kusano et al.  found that among women with a BMI under 25, those with a cup size of D or larger had a 1.8 times higher risk of breast cancer than those with a cup size of A or smaller."
So I went to the font of all wisdom and looked up cup size to volume and found that it is a complete mess (and apparently manufacturers have vanity sizes). This study used strap size as a proxy for BMI, so I will too. Picking a few sizes with the same strap size one finds: the ratio of estimated volume for a 34D to a 34A is about 1.9, 36D to 36A is about 1.8, 38D to 38A is about 1.8. So maybe my intuition is OK. If size matters most, then how you got the size should have the biggest bearing on what kinds of cancer you’re at risk for and when.
Their Figure 1 is pretty dramatic. But they warn
... the shared relationships between breast size and breast cancer at these three regions are not strong enough to account for the possible epidemiological connection that has been reported elsewhere between breast size and breast cancer
Maybe their volume estimation isn't clean enough. Or maybe the naive intuition isn't enough of the story.
BTW, increased BMI increases cancer risk in parts of the body that don’t necessarily increase in size/cell count proportionately. Cancer stories are complicated.
 Kusano et al. Int J Cancer 2006, 118:2031-2034