In common English, people use "can" and "may" interchangeably. But traditionally, "can" implies ability (you are physically able to open the door and walk away), whereas "may" implies permission. If the person asks "may I leave," then the safest positive answer is "yes you may" (or just yes).
"Must" implies something beyond permission. For example, you have permission to leave your own house, but if there's a terrible storm outside, someone might say "you mustn't leave! It's too dangerous!"
Note: In my opinion, "must" is becoming a bit old-fashioned in common use, especially in the US.