Kant discusses this within the first paragraphs of his Critique of the Power of Judgement.
This is really a matter of definitions here, and as all too often, if you take Kant's definitions, the outcomes are analytically derived from them:
In order to decide whether or not something is beautiful, we do not
relate the representation by means of understanding to the object for
cognition, but rather relate it by means of the imagination (perhaps
combined with the understanding) to the subject and its feeling of
pleasure or displeasure. (5:203)
The judgement of beauty therefore links a representation [Vorstellung] to the subjective feeling it evokes. He writes regarding interest:
The satisfaction that we combine with the representation of the exis-
tence of an object is called interest. (5:204)
That means that while there is a link between subjective feeling ('satisfaction') and a representation, it is the representation of the existance of the object. As he continues to argue in §2, beauty does not need the existence of the object of the representation that evokes the pleasure at all, but solely refers to the representation itself. The feeling of beauty does not need any objectivity (or even 'intentionality') thought in the representation.
Therefore, it is desinterested in this sense. And he even clarifies in a footnote on 5:205:
A judgment on an object of satisfaction can be entirely disinterested yet still
very interesting, i.e., it is not grounded on any interest but it produces an
interest; all pure moral judgments are like this. But the pure judgment of
taste does not in itself even ground any interest. Only in society does it
become interesting to have taste, the reason for which will be indicated in
That simply is because satisfaction (and therefore interest) is thought as linked to the existance of an object (as contrasted to its representation) and the whole discourse is in the realm of existing objects, while beauty only refers to representations, no matter if they actually exist or not. The existence of the object would not change anything. Therefore, it is desinterested, because there is no satisfaction linked to the existence of its object whatsoever.
Misunderstandings mainly occur because of mixing the feeling of beauty with the satisfaction in interest as he defines it.