There are some specialized terms here that you're unlikely to encounter outside of an academic philosophy paper, but there's nothing complex about the meanings of any of the individual terms. Once you know what the words mean, it all makes sense.

>eliminativist

Eliminativist claims in philosophy are claims that deny the existence of some class of entities. You can be eliminativist about all sorts of things - numbers, objective morals, countries, tables and chairs, etc.

>qualia

First-person conscious experiences. Pain is a qualia. The way the color blue looks, as opposed to say the color red or green, is a qualia. The sensation of hot or cold is a qualia.

When someone stubs their toe and says "ow", you can infer that they're in pain based on their behavior and your knowledge of how pain works, but you can't actually feel or directly observe their pain. That's the "first-person" part.

>phenomenal consciousness

A synonym for "qualia", because some philosophers started to feel like the word "qualia" had too much historical baggage, so they needed to come up with a new term.

>introspective illusion

Exactly what it says on the tin. An illusion (meaning, an impression that something is real, when it is in fact not) generated by introspection.

So, putting it all together:

>illusionism

Illusionism about consciousness is the thesis that phenomenal consciousness is not real. So, to give a specific example, an illusionist would be committed to the thesis that pain is not real. As a corollary, no one has ever felt pain before, because there is no such thing as pain. People have been under the illusion that they feel pain, but they actually don't.