Many philosophers today, including materialists, are heirs to Cartesian metaphysics, the same metaphysics that brought us the mind-body problem and the problem of qualia. The difficulties the author has in mind are very much bound up with this Cartesian (and Galilean) legacy.
The idea that material beings can be conscious is often treated like it's some new and shocking claim. The reason for that is, again, the broadly Cartesian heritage at work. For Descartes, it takes an immaterial "res cogitans" to be conscious. The material body, as a desiccated "res extensa", is incapable of functioning as a substrate for anything we might call "consciousness". And because Descartes holds that only human beings possess minds understood as "res cogitans", only human beings can possess consciousness. Furthermore, "res extensa" lacks sensory qualities like color or sound. As a result, these qualities are located in the "res cogitans" as immaterial "qualia". Materialism does not escape the Cartesian paradigm. Instead, it denies the "res cogitans" and tries to locate what was ascribed to the "res cogitans" back in the "res extensa" while maintaining many of the metaphysical and methodological tenets of Cartesianism. This is where the problem of qualia occurs. The problem is insurmountable as long as the Cartesian suppositions are maintained. No amount of "keeping at it" with the current methods will ever resolve the issue. (See Nagel for more.)
For what it's worth, Aristotelian metaphysics encounters no such difficulties because it does not adhere to the dualism of Cartesian metaphysics (though sadly, few philosophers understand Aristotelian metaphysics; there seems to be a resurgence of interest, however). Indeed, for Aristotle, "consciousness" (though he does not use this term) is part and parcel of what it means to be an animal, human or otherwise. (See Jaworski and Feser for more.)
"It’s true that people can make all sorts of mistakes about what is going on when they have experience, but none of them threaten the fundamental sense in which we know exactly what experience is just in having it."
If there was any doubt about the Cartesian flavor of the author's reasoning, this statement should have dispelled it.
"Members of the first group remain unshaken in their belief that consciousness exists, and conclude that there must be some sort of nonphysical stuff: They tend to become 'dualists.' Members of the second group, passionately committed to the idea that everything is physical, make the most extraordinary move that has ever been made in the history of human thought. They deny the existence of consciousness: They become 'eliminativists.'"
Ah, but the eliminativists haven't escaped the legacy of Cartesian dualism either! They've eliminated not only the "res cogitans", but the facts of experience (e.g., "qualia") altogether! It's a notoriously incoherent position.
Strawson says he's a panpsychic physicalist, but that falls into the dualist camp once you get down to it. He offers no third way between Cartesian dualism and eliminativism.