The stupidest thing about the call was that you have to stop the Saints whether you punt or fail to convert on 4th. The only reason to punt was that the Saints would have been on the edge of FG range after a failed 4th down conversion, so they could have made it a 6 point game instead of a three point game.
Even if the Eagles only convert 10% of the time, it's a good call, since a successful conversion almost always gives them a FG attempt and will sometimes result in a game winning TD.
Punting forces them to 1) stop the Saints, 2) drive the ball into FG range with under a minute left and no timeouts, 3) make the winning kick. Their chances of doing all three were probably less than their chances of converting the 4th and 15, and even if they don't convert, they still get a chance to do all those things, albeit from worse field position most likely.
Just a terrible decision with the game on the line.