To answer the title question we need to know what we mean by 'explain' and how it differs from 'explain away.'
1. An obvious point to start with is that only that which exists, or that which is the case, can be explained. One who explains the phenomenon of the tides in terms of the gravitational effect of the moon presupposes that the phenomenon of the tides is a genuine phenomenon. One cannot explain the nonexistent for the simple reason that it is not there to be explained. One cannot explain why unicorns run faster that gazelles for the simple reason that there is no such explanandum. So if consciousness is to be explained, it must exist.
2. A second point, equal in obviousness unto the first, is that a decent explanation cannot issue in the elimination of the explanandum, that which is to be explained. You cannot explain beliefs and desires by saying that there are no beliefs and desires. A successful explanation cannot be eliminativist. It cannot 'explain away' the explanandum. To explain is not to explain away.
3. Summing up (1) and (2): the very project of explanation presupposes the existence of the explanandum, and success in explanation cannot result in the elimination of the explanandum.
4. Daniel Dennett points out that there can be no explanation without a certain 'leaving out': "Leaving something out is not a feature of failed explanations, but of successful explanations." (Consciousness Explained, 1991, p. 454.) Thus if I explain lightning as an atmospheric electrical discharge, I leave out the appearing of the lightning to lay bare its reality. That lightning appears in such-and-such a way is irrelevant: I want to know what it is in reality, what it is in nature apart from any observer. The scientist aims to get beyond the phenomenology to the underlying reality.
5. It follows that if consciousness is to be explained, it must be reduced to, or identified with, something else that is observer-independent. Dennett puts this by saying that "Only a theory that explained conscious events in terms of unconscious events could explain consciousness at all." (454) For example, if your explanation of pain in terms of C-fibers and Delta A-fibers (or whatever) still contains the unreduced term 'pain,' then no satisfactory explanation has been achieved. There cannot be a "magic moment" in the explanation when a "miracle occurs" and unconscious events become conscious. (455)
6. Now if a successful explanation must explain conscious events in terms of unconscious events, then I hope I will be forgiven for concluding that consciousness CANNOT be explained. For, as I made clear in #2 above, a successful explanation cannot issue in the elimination of that which is to be explained. In the case of the lightning, there is a reduction but not an elimination: lightning is reduced to its observer-independent reality as electrical discharge.
Now suppose you try the same operation with the sensory qualia experienced when one observes lightning: the FLASH, the JAGGED LINE in the sky, followed by the CLAP of thunder, etc. You try to separate the subjective appearance from the observer-independent reality. But then you notice something: reality and appearance of a sensory quale coincide. Esse est percipi. The being of the quale is identical to its appearing. This is what John Searle means when he speaks of the "first person ontology" of mental data.
7. It follows from #6 that if one were to explain the conscious event in terms of unconscious events as Dennett recommends, the explanation would fail: it would violate the strictures laid down in #2 above. The upshot would be an elimination of the datum to be explained rather than an explanation of it. To reiterate the obvious, a successful explanation cannot consign the explanandum to oblivion. It must explain it, not explain it away.
8. I conclude that consciousness cannot be explained, given Dennett's demand that a successful explanation of consciousness must be in terms of unconscious events. What he wants is a reduction to the physical. He wants that because he is convinced that only the physical exists. But in the case of consciousness, such a reduction must needs be an elimination.
9. To my claim that consciousness cannot be explained, Dennett has a response: "But why should consciousness be the only thing that cannot be explained? Solids and liquids and gases can be explained in terms of things that are not solids, and liquids, and gases. . . . The illusion that consciousness is the exception comes about, I suspect, because of a failure to understand this general feature of successful explanation." (455)
Dennett's reasoning here is astonishingly weak because blatantly question-begging. He is arguing:
A. It is a general feature of all successful explanations that F items be explained in terms of non-F items
B. Conscious items can be explained
C. Conscious items can be explained in terms of nonconscious items.
(B) cannot be asserted given what I said in #6 and #7. I run the argument in reverse, arguing from the negation of (C) to the negation of (B): conscious items such as pains are irreducible.
10. Recall from #4 that Dennett said that successful explanations must leave something out. But in the case of a conscious item like a pain, what is left out when we explain it is precisely what we needed to explain! For what is left out is precisely the sensory quale, the felt pain, the Feiglian "raw feel,' the Nagelian "what it is like."
11. Amazingly, on p. 455 he retracts what he said on the previous page about successful explanations having to leave something out. He now writes:
Thinking, mistakenly, that the explanation leaves something out, we
think to save what otherwise would be lost by putting it back into
the observer as a quale -- or some other "intrinsically" wonderful
property. The psyche becomes the protective skirt under which all
those beloved kittens can hide. There may be motives for thinking
that consciousness cannot be explained, but, I hope I have shown,
there are good reasons for thinking it can. (455)
Do you see how Dennett is contradicting himself? On p. 454 he states that a successful explanation must leave something out, which seems plausible enough. Then he half-realizes that this spells trouble for his explanation of consciousness -- since what is left out when we explain consciousness in unconscious terms is precisely the explanandum, consciousness itself! So he backpedals and implies that nothing has been left out, and suggests that someone who affirms the irreducibility of qualia is like a lady who hides her 'kwalia kitties' under her skirt where no mean neuroscientist dare stick his nose.
The whole passage is a tissue of confusion wrapped in a rhetorical trick. And that is the way his big book ends: on a contradictory note. A big fat load of scientistic sophistry.
12. To sum up. A successful explanation cannot eliminate the explanandum. That is nonnegotiable. So if we agree with Dennett that a successful explanation must leave something out, namely, our epistemic access to what is to be explained, then we ought to conclude that consciousness cannot be explained.