'Reasons' are typically introduced as 'facts that count in favour' of an action or attitude. But this invites confusion, insofar as both fittingness and fortunateness considerations may be taken to 'count in favour' in some sense. For example, there's a sense in which the truth of a belief counts in its favour (it renders the belief fitting or correct), and another sense in which the benefits you'd receive from having a certain belief count in its favour (it makes the belief fortunate or good to have). But it would be messy and unhelpful to call both of these 'reasons for belief'. Better to talk exclusively in terms of fittingness reasons -- and then when we want to note the fortunateness of a belief, we may do so by saying that it would be fitting to desire that one have the belief.
Broad considerations of theoretical elegance aside, this move may also be motivated by reflecting on the concept of a reason. For example, one may think that reasons must be followable: if p is a reason for you to φ, then it must be possible to φ for that reason. That is, there's a close connection between reasons and reasoning. In general, if p a reason for φ-ing then there is a sound chain of reasoning that would lead one from the premise that p, to the conclusion to φ. (Though it's worth noting that one may rationally respond to a reason, and act "for" that reason, without such explicit deliberation.) Put yet another way: reasons determine what would be rational in light of all the facts. But just because you'd be rewarded for (say) believing that the earth is flat, doesn't make this a rational belief. It merely makes it rational to desire the belief, or to act so as to cause yourself to acquire this irrational belief. These truisms about rationality should be reflected in the claims we make about reasons. Hence, the value of a belief is not a reason for belief; it is merely a reason for desiring or acting to acquire the belief.
So we can reach this conclusion by starting with 'reasons' talk. But it's easier, and I think clearer, to instead start with 'fitting attitudes' talk. For example, the notion of 'desirability' strikes me as much clearer and less ambiguous than the notion of a 'reason for desire'. Suppose that an evil demon threatens to harm you unless you form a desire for pain. This clearly doesn't make pain desirable (i.e. fitting to desire). What it makes fitting to desire is instead the following: that you desire pain. From here we can derive the notion of a 'reason for desire' as a feature in virtue of which an object is fitting to desire (in short, a 'desirability characteristic'). This enables us to analyze value in terms of desirability without having to worry about the so-called "Wrong Kinds of Reasons".
Having thus clarified the framework, you might be wondering: why bother with all this? How does taking fittingness as our normative primitive help us more than starting with value, or some more general ought?
(1) It is more elegant and parsimonious, in that a single notion of 'fittingness', applied to various sites of rational agency, can underlie and unify such superficially disparate concepts as 'the good' (what's fitting to desire) and 'the right' (what's fitting to choose or do).
(2) Analyzing value in terms of fitting attitudes clarifies the normative significance of evaluative claims to us as agents. (Consider: what would it matter to call something 'good' if that didn't have any implications for how we should react to it?)
(3) By focusing in on the 'ought' of fittingness (rather than some more general or ambiguous deontic notion), we can discipline our normative theorizing. For example, some Global Consequentialists want to apply deontic language more broadly, and so say that the best climate is also the 'right' climate, or the climate that 'ought' to be. But it is unclear what further claim one might be making here, besides the evaluative one that a certain climate is best (most desirable) among the relevant options. The fitting attitudes approach provides us with a test to confirm that these aren't substantive new claims: see whether they imply any new claims about what rational responses would be fitting on our part as agents.
(4) We can draw more careful and fine-grained distinctions, e.g. between bringing about some act or attitude, and the act or attitude itself. Combined with the 'fittingness' conception of reasons, this avoids normative ambiguity in cases where some fitting attitude would be unfortunate (or vice versa). Consider again the example where you'll be rewarded if you come to believe that the world is flat. In deciding whether to take a pill to acquire this belief, it's not as though there are two competing kinds of reasons to consider: 'practical' ones that count in favour of bringing about the belief, and 'epistemic' ones that count against this very same thing. 'Bringing about the belief' is an action, which the epistemic reasons don't speak to (or count against) in the slightest.
(5) Indiscriminate theories have trouble explaining why rational agents can't form beneficial beliefs at will (or beat Kavka's toxin puzzle, etc.). If you think there's a normative "reason" supporting any beneficial attitude, why can't a rational agent form the attitude (e.g. believe the world is flat, or desire pain, or intend to drink the toxin) for just that reason? The fittingness approach answers that we can't rationally do it because there is no reason, in these cases, for those attitudes themselves. There's nothing there to properly engage one's rational capacities for (say) belief-formation. So a rational agent won't be able to form a belief on that basis (much as they might wish they could). They might rationally cause themselves to acquire the irrational belief, e.g. by taking a pill, but as we've seen that's a different process than straight-out believing.