VincentYu has it right. They 1) used biases that had previously been shown to be uncorrelated with intelligence and 2) checked and confirmed that the biases actually were uncorrelated with intelligence in their sample.

Having read the paper, I don't think they can, and they admit this to some extent. What they can prove is that in the small selected set of biases they chose, the smart are incorrect; but to show that the smart are miscalibrated in general about their resistance to biases, one would need to do a lot more work. (There are scores of biases, after all.)

They studied the bias blind spots of cognitive biases that show no correlation with cognitive ability.

The finding that the bias blind spot is more apparent among the more cognitively sophisticated individuals is contrary to much of the rest of the heuristics and biases literature where most biases are negatively correlated (or at least independent) with cognitive abilities [...] Given the rarity of the result, it is important to consider alternative explanations for it, and an obvious one suggests itself. Perhaps more cognitively sophisticated individuals actually are less likely to display these classic cognitive biases. Such a finding would in fact cause a positive correlation if more cognitively able individuals correctly perceived that they were less prone to bias. This alternative explanation has diminished likelihood, however, when we reflect on the fact that these cognitive biases have been found in another study to be independent of cognitive ability (see Stanovich & West, 2008b, Table 1).

We explored the obvious explanation for the indications of a positive correlation between cognitive ability and the magnitude of the bias blind spot in our data. That explanation is the not unreasonable one that more cognitively sophisticated people might indeed show lower cognitive biases—so that it would be correct for them to view themselves as less biased than their peers. However, as the analyses in Tables 3 and 5 indicate, we found very little evidence that these classic biases were attenuated by cognitive ability. More intelligent people were not actually less biased—a finding that would have justified their displaying a larger bias blind spot.

ETA: On further thought, the lack of correlation merely simplifies the study, and is not a necessary assumption – for each subject, they administered tests on the selected cognitive biases, obtained several measures of cognitive ability/sophistication (CRT, SAT, etc.), and asked for subjective assessments on the extent of the biases:

To what extent do you believe that you are likely to commit base-rate neglect?

To what extent do you believe that the average JMU student is likely to commit base-rate neglect?

They used the subjective assessments along with the results of the administered tests to derive the extent of bias blind spots, independent of the measures of cognitive ability. These are then combined in a standard correlation study, and your hypothesis can be tested.

If that were true, and if the relation between measures of "cognitive sophistication" and of bias were continuous, then by the intermediate value theorem there would be some level of "cognitive sophistication" which minimized overall bias. Of course, that might be an unstable point; anyone smarter than that tends to fall into a bias-blind-spot attractor and anyone less smart than that tends to fall into an object-level bias attractor.