I think to understand Dennett's position better it may be worth starting with epistemology (naturalism) rather than metaphysics (materialism). It's also worth noting that it can end up as a more modest position than it seems at first, by virtue of just what we're rejecting when we say we're rejecting qualia.
Let's say we start with a naturalist methodology - that is, a position that says that philosophy should be committed to the same methodologies as science, broadly construed. Naturalism doesn't *necessarily* lead to materialism. You could start with a hypothesis that there are ghosts, formulate a bunch of secondary hypotheses about what would happen if there were or were not ghosts, go into some haunted houses, see how the hypotheses hold out, then let other people reproduce them.
All of this involves communicating public information. Now, if you have a private experience with the supernatural, unless you can convert it into publically verifiable/communicable information, it's not science.
So let's get past ghosts and move on to qualia. It's important to note that "qualia" is a narrower category than "experience," in particular, it's what's left of experience *when you take out all the public facts about it.* Think of the archetypal case, Mary the color scientist, a congenitally colorblind researcher who knows everything there is to know about color (what wavelength red is, the cultural associations of red, how seeing red affects humans and bulls, whatever) except what it looks like. Then she gets eye surgery and can see color, learning "what red looks like?" What kind of knowledge is this, if any?
From the perspective of methodological naturalism, because the extra content of the quale of redness - what's left after we subtract every externally observable fact - by definition can't be communicated, it's (by definition) meaningless to speak of it. Philosophy is, after all (or at least on this conception), a set of public speech acts.
Now I think there are a number of different ways you can respond to this. One is to say it's not that weird, Dennet isn't denying consciousness, he's denying a supposed nonfalsifiable extra ingredient to consciousness, or, rather, that we could coherently have conversations about such a thing. Another is to say, wait, everyone is sitting around insisting that they have this extra thing going on with the redness of red above and beyond observable facts, and surely *that* (people's tendency to do this) is an externally observable fact that we need to better account for, possibly by saying "yeah there likely is SOMETHING else even if it's really hard to say what it is." (It's been a while since I've read Dennett so I forget how well he does attempt to explain this.) A third response would be to reject methodological naturalism and say that philosophy is something you can in fact do within the confines of your own skull, no public communication with other actors required.
I don't think any of these options really solve the Hard Problem, it all seems pretty spooky to me even after reading a great deal on it. But Dennett starts with a defensible (even if not my favorite) conception of what philosophy is, and then every step he takes from there to eliminative materialism is likewise very reasonable.