Why is zai used here?
to show the equivalent of verb + ing.
It is used when you are in the middle of doing something, so as Riley says you use the present continuous
在 here is showing the equivalent of verb + ing (I think )
Why my answer:"what do you write" was rejected?
寫著 could also be "writing" the English is problematic. the 在 is indicating the action is presently taking place. it would be clearer if something were added to indicate ongoing action in the English translation.
The zai threw me because I thought it was the subject of the question - "Are you writing something (now)? How would you ask this question?