In general, "naturalizing" X, for philosophers, meaning looking at how the processes of X occur in nature, and developing a theory of it based on those observations rather than reasoning about it in the abstract. Quine's endorsement of "naturalized epistemology" was therefore an effort to refocus philosophical analyses of knowledge towards how knowledge is actually produced in the world (i.e. in nature). This would move epistemology away from the study of what kinds of reasons for believing a sentence offer adequate justification for it. It would move epistemology towards the study of human psychology (and how individuals acquire knowledge) and natural science (and thus how we collectively acquire knowledge).
Not being a Habermas expert, I don't know whether more is built into his phrase "moment of unconditionality" but I strongly suspect that "naturalizing reason" here means looking at how human beings in fact from evidence to conclusions, and indeed replacing logic with that. I suspect that "dropping the moment of unconditionality" means removing the abstraction from the analysis of reason, and shifting it towards reasoning as it actually occurs, embedded in real conditions.
Now, in both cases, there are serious questions about whether a naturalized inquiry could ever replace a traditional one, even as it seems right that there is a danger in analyzing processes entirely from the armchair, without engaging with how they happen in in fact. But the process of integrating those two approaches has indeed characterized a good deal of Anglophone philosophy over the past couple decades. But, I think that's what Habermas's critic is saying above.