Many (if not most) airports are open 24 hours, although the terminal may not be.

Just think about if you were on a flight that was scheduled to arrive at 11 PM, and the airport shut down at midnight, but because of WX or ATC you are running 3 hours late, and you have to be in the office the next morning. Would you want to A) land at your airport at 2 AM (and get to your home or hotel for a few hours rest), or B) land at some other airport and have a 250-300 mile bus ride?

Many late flights you hear are late commercial flights arriving or express/freight flights (FedEx, UPS, etc...). Most of these fly overnight. (Overnight packages! )