I think 'Greg' and 'Christine' meet at the airport by chance. But how "met" is possible in this sentence? Excuse me, Are you Mr. Greg? Yes, and You're Christine? Bingo. I can't believe we met here. How are you doing?
Do you mean why "met" instead of "meet"? This is because the action of meeting is considered to be over as soon as it has happened, so unless it is to happen in the future, the past is always used. (Unless another past verbs are involved, as in "we happened to meet", or "how did you meet her", etc.)
If you speak at the instant of meeting you can say "We meet again", but after that it is "met".