My first reaction: I bet this is correlated with age.
Come on, guys, read the paper. The main point of the paper is that smokers earn less than nonsmokers, and the authors are looking at factors that contribute to this wage gap. The headline about quitters earning more than smokers and never-smokers is a nice attention-grabber, but it's a tangential finding; it's not news either, since the authors mention that this has been observed before.
As for what's behind the claim that former smokers earn more than anybody else, look at the raw data. Even if you are skeptical about the use of linear regression models to 'explain' social behavior (and I count myself in that camp), you can still check out Table 2 on page 43, which reports simple averages by group. Former smokers are older than any other group, so it makes sense for them to earn more. Why would they be older? The group of former smokers includes those who quit because of health reasons, typically after years of accumulation of smoking's side effects.
The authors' assertion that "former smokers have attributes that are more highly rewarded in the labor market than either current smokers or never smokers" is pretty bogus. Right, the attribute in question is called AGE. 'Ability to quit', as the headline claims, has nothing to do with the higher rewards.