Adding hops at flame out literally means when you turn off your heat source at the end of your boil. I usually take out my hops after I cool, but I suppose theirs not much of an advantage either way unless they get in the way of an immersion cooler. If they are a flame out addition you want to keep them in there until it goes in the fermenter.
Like you steep specialty grains to wash off sugars, you steep hops to wash off essential oils that add to aroma and flavor. You'll definitely want to leave them in for the 30 minutes or so it takes to cool the batch down, to make sure you get all of the goodness they have to offer.
When I have recipie that calls for a flameout hops addition, I generally put them in the last 5 minutes of the boil. But, as stated above, the recipie actually calls for hops be left in while the wort is cooling. I strain mt wort after cooling through a stainless steel seive into the fermenter to remove the hops and trub. That means the hops are all left in the 20 minutes it takes to cool to fermenter temps, plus the recipie boil time.