I think there are different senses of ‘illusion’ that need to be distinguished here. One sense is that of a non-veridical experience: I have the illusion of there being a black cat in the room if I have an experience of seeing a black cat in the room when there isn’t, in fact, a black cat in the room. In this sense, saying ‘conscious experience is an illusion’ is indeed nonsense: I can’t have a non-veridical experience of conscious experience—indeed, I don’t have any experience of experience at all, the experience is all there is!
But that experience can be systematically non-veridical, and thus, be illusory in a different sense. This sense concerns false conclusions that are only apparently implied by some data—in a literal sense, for instance, if I have a study of some illness indicating the effectiveness of a treatment which turns out to have been a statistical fluke (in which case I could rightfully say that its efficacy turned out to be illusory), or in a more figurative sense, if some items of knowledge—perceptions, memories, and the like—strongly imply something to be true that actually isn’t, as in when I think that the hairs on the cushion imply there was a black cat in the room, but there actually wasn’t.
So illusionism, in my opinion, means that experience is systematically non-veridical in the sense that it doesn’t have the properties that we normally, and perhaps unavoidably, associate with it, and thus, the word ‘experience’ does not actually refer to anything real in the world, but merely our misguided beliefs. For instance, experience is often claimed to be ineffable—that is, I could never explain to a congenitally blind person ‘what it’s like’ to see the color red. In principle, however, that could be wrong: it could merely be really, really hard, involving the transfer of quantities of information that we can’t fathom transferring via language, or transferring that information in a way that we haven’t yet thought of. (For instance, there are people that claim to be able to visualize four-dimensional objects, without obviously ever having seen one; thus, the capacity of having that experience must be something that’s transferable without the need of actually having that experience, and that transfer presumably was enabled only by our development of the requisite mathematics.)
It’s this sense in which consciousness may turn out to be illusory: the properties we consider it to have may simply turn out not to apply to it, and hence, the term fails to refer to anything in particular—i. e. there’s nothing of the sort of what we believe ‘consciousness’ is in the world.