Learn Italian: Mi grattugio questo formaggio, che ho fame. - I am grating myself this cheese, because I am hungry.

In informal spoken Italian, che is frequently used as a quick, casual replacement for perché, poiché, or siccome (because/since). It acts as a coordinating conjunction to justify the first part of the sentence. This is exactly what is meant by the explanatory che (or che polivalente) taught in this lesson.

What does the mi add to grattugio? Could I just say Grattugio questo formaggio?

You absolutely could just say grattugio. Adding mi is an example of the "ethical dative" (or reflexive of advantage). It emphasizes that you are doing the action for your own benefit, comfort, or enjoyment. It translates roughly to "I am grating myself this cheese" and is extremely common in spoken Italian, especially when talking about food (e.g., mi mangio un panino).

Could I replace che with cui or il quale in this sentence?

No. While che is often a relative pronoun (meaning "who" or "which"), in this specific sentence it is acting as a conjunction (meaning "because"). Since it is connecting two clauses to show cause and effect, rather than referring back to the cheese, relative pronouns like cui or il quale would make no sense here.

How would I rewrite this sentence in a standard or formal register?