why is it in this case, te is used instead of tu?
Ex: "il te lit un menu" instead of "il tu lit un menu"
"tu" is used only when it is subject of the verb. When used as a complement (here indirect complement), you use "te".
Why does "Il lui lit un menu" not work?
Because "lui" is 3rd person singular, while "vous" is 2nd person plural (polite "you" or several persons)
Why does Il te lit un menu not work when the sentence is "a" menu?