Two things you can say
...and one of them is wrong. (More from JFK).
"You must be feeling really frustrated."
What a great thing for a gate agent to say to a frustrated traveler. I saw it used three times in ten minutes, and it worked every time. It enabled the agent to get on the same side of the conversation, it allowed the customer to let off some steam and got both sides moving.
On the other hand,
"Poor planning on your part does not constitute an emergency on my part..."
This is true, of course, unless your goal is to make the person happy, or, at the very least, get rid of them. There were all sorts of clueless people at the airport today, cutting lines, yelling, getting angry just because they didn't leave enough time. Not the airline's fault, that's for sure.
Yet the best way to handle the situation is not to persuade, convince or bully the person into admitting that they were wrong. No reason to teach these people a lesson, because they're not going to learn a lesson anyway.