Just curious, but why is the "che" necessary in Italian? Does it have something to do with this verb tense, and without it, one would write Volgio che tu la vedere.
It's just how you express sentences like these in Italian (and Spanish and probably other Romance languages). "I want you to see it" = "I want that you see(subjunctive) it"
It's true that this construction sounds unnatural to modern English speakers, but in the past one would have run into the (admittedly obscure) construction "I want that you should see it," a subjunctive form that has fallen out of use. But it's worth noting that English has sometimes had a place for "that" in the present subjunctive.
It seems to help "reset" who the next verb is conjugated for. It doesn't work to say, "Voglio veda" but does to say, "voglio che veda." The first example looks like I tried to say "I would like to see" but didn't use the infinitive form as I show have for "vedere."
I know this response is a bit late, but you usually use the subjunctive in clauses that have different subjects, For example, I want to see it (voglio vederla) vs this sentence in the lesson.