If the sentence were, "They do not want to eat meat.", then the infinitive "mangiare" could have been used after the verb "volere" (to want which has the form "vogliono" to be used with the 3rd person plural pronoun which is not in the sentence, because Italian does not require the subject to be there. The subject is understood to be "they" or "Loro")
However, "They do not want" or " Non vogliono" is followed by a clause "that you eat meat." or "che tu mangi carne". So "mangi" has a different subject than "vogliono" and must be conjugated to match its own subject which is "tu" (2nd person singular pronoun, their familiar version of you). In English we often drop "that" in this situation, but the whole clause still remains as a direct object of the verb want.
What don't they want? They do not want you to eat meat The reality is that this is originally the sentence "They do not want that you eat meat." and the clause is originally in the subjunctive mood. In Italian, it is also in the subjunctive ("congiuntivo"), but their subjunctive for "tu" is exactly the same form that is used for the present tense in indicative mood (Most of our language is in indicative mood). The confusion comes when we drop "che" or "that" and change the verb form to infinitive, but in Italian they do not do this. In English we often avoid using the subjunctive mood and it is really less used than in other languages, but it is alive and well-used in Italian.
This is showing the subjunctive case in the clause. What do they want? "that you eat meat" (subjunctive) more commonly nowadays in English "you to eat meat". Either form is correct, but the first corresponds to the Spanish version which requires the subjunctive form.
I wrote 'they do not want you to eat that meat' which was marked as incorrect. I used 'that' for the 'che', but the correct answer was 'they do not want you to eat meat' I thought I understood all the meanings of 'che' but does it also mean 'to' as well? I'm a little confused why it's needed in this sentence and dispite reading all the comments I'm still not clear... Help please