Flame out is a term used to describe additions made to the kettle after the heat has been turn off of the kettle. If you are talking about hops it is usually the time for getting a big addition of aroma hops into the kettle.
Other things like orange peels, coriander, chocolate nibs and goats can be made in this addition as well. The termination of the rapid boil slows the rate of precious volatile oils from all these additions leaving the kettle, which could be bad in the case of goat, because you might want to drive those aromas out.
However, there is still considerable heat in the wort, that you are able to nearly sterilize anything you put in the wort at this point.
(A good thing in the case of goat additions. You certainly want sterile goat in your wort).
As opposed to adding spices, for example, in a secondary; where I have had occasional infection occur because not all spices are that clean as they often sit around your kitchen for ages before you use them.
Flame out is that time frame between turning of the heat and starting the chill. In some pro applications, this time frame is also when the kettle wort starts to get whirlpooled to draw break material to the center prior to run off. So the flame out can last several minutes to just seconds depending on the scale of your operation and other post boil activities (additions vs whirlpool).