Breakdown of تو چای نمیخواهی؟ چرا، میخواهم.
تو
you (singular)
چایchâi
tea
چراcherâ
why
خواستنkhâstan
to want / to ask (request from someone)
Questions & Answers about تو چای نمیخواهی؟ چرا، میخواهم.
I thought چرا meant "why". Why is it translated as "yes" here?
You are right that چرا usually means "why". However, when you answer a negative question (like "Don't you want tea?"), you use چرا to say "yes, I actually do". If you used the regular word for yes (بله), it would sound like you are agreeing with the negative statement!
How is the negative verb نمیخواهی put together?
It comes from the present tense of the verb "to want". The prefix می shows it is a present action, and adding نـ to the front makes it negative (نمی). The ending ـی shows it means "you". So, it means "you do not want".
The English translation says "I do want it", but where is the word for "it" in Farsi?
There isn't one! In Farsi, when the object you are talking about (the tea) is already obvious, you can just drop it entirely in your reply. You only need to say میخواهم (I want).
Why does the question include the pronoun تو (you) but the answer drops the pronoun for "I"?
Because Farsi verb endings tell you exactly who is doing the action, pronouns like تو (you) and من (I) are optional. The speaker included تو in the question to be direct or polite, but the answerer just relies on the ـَم ending on میخواهم to show that "I" am the one wanting tea.