Learn Italian: Non è un principiante, anzi, sa di essere qualificato. - He is not a beginner; on the contrary, he knows he is qualified.

In Italian, when verbs of knowing, believing, or saying (like sapere) have the same subject in both the main clause and the dependent clause (he knows + he is), you use the implicit construction: di + infinitive. So instead of using che and a conjugated verb, you use sa di essere qualificato.

How exactly does anzi work in this sentence?

Anzi means 'on the contrary' or 'rather'. It is used to contradict or correct a negative statement that came just before it. Here, it refutes the idea of him being a beginner to introduce the stronger affirmative fact that he is actually highly qualified.

Would qualificato change if the subject were a woman?

Yes. Even though essere is in its infinitive form here, the adjective qualificato still must agree with the implicit subject it refers to. If the subject were female, the sentence would be: Non è una principiante, anzi, sa di essere qualificata.

Could we use invece instead of anzi here?

While invece (instead / on the other hand) also shows contrast, anzi is a much better fit. Anzi specifically refutes the previous negative statement to step up to a stronger, opposing truth, whereas invece usually just points out a difference between two separate situations.