A: Hey man! Are you free tonight?
B: Sorry dude! I'm very busy tonight.
A: Oh! What are you busy about?
- Is this conversation grammatical?
Okay, but "busy about" doesn't sound English.
What are you doing that's so important?
What are you going to be doing?
Or just change the subject, talk about something else. The man who says he's going to be busy is probably being evasive, possibly ducking an unwelcome invitation, and won't like to be interrogated about his plans.
But this note is cultural, not linguistic.
To me, using man and dude in this way sounds more like 1970 than 2007. I could be wrong; both these expressions are American, and were never used much in Canada.