You are sitting by a friend in an airport and want to send them a great video of a cat.

  You$ airpaste cat < funny-cat.m4v

  Them$ airpaste cat > the-stupid-video-i-have-to-watch-now.m4v

You don't have to dig out your IP address. You also don't have to toss it up to a server with some awful upload speed, it all stays local.

The name suggests the author was thinking of Apple's AirDrop system which does sort of the same thing. You can share content to an iOS device or Mac without leaving your local network. You pick a recipient from the available, acceptable, recipients and if they accept the content transfers. The usual way of filtering recipients is to only see recipients who are in your contacts.

I used some OS X APIs like this to have test machines and a test controller machine find each other. That was a while ago so I don't remember the details, but maybe it was built on the same APIs.