What does "out round" mean in NFL?

My interpretation is that the fake psychic character is using this as a situation to show his knowledge (or psychic ability) by predicting the play that the offense was about to run. It looks to me that he says "out round" not to the other players, but to his partner, and then celebrates the result of the play as if his prediction were correct.

Now, what would "out round" mean here? As Ram commented, it's similar enough to "out pattern" that it sounds like a route for a receiver to run.

Unless someone knows of a specific meaning for the phrase, my suspicion is that it is simply a throwaway line that didn't mean anything in particular, but sounds like a plausible NFL play.

You mean out route? Same thing as down & out or a jet route. It's a pattern that receivers run, starting by running straight down the field toward the EZ. After a certain number of steps, he'll make a hard cut (90 degrees) toward one of the sidelines.

This kind of route is good for forcing 1v1 coverage. It's a great way to move down the field and stop the clock because the receiver will be able to hop out of bounds after catching the ball--for this reason, you'll often see the route run under 2 minutes.