The functions of auto-thrust include thrust reduction during a flare-out. What is flare out and why during flare out will the auto-thrust reduce thrust?
The proper term is "flare" not "flare-out".
"Flare" is a term used to describe the act of raising the nose of an aircraft in order to increase lift and stop descending in order to land on a runway smoothly.
Whether the pilot, or Auto Thrust does it, a reduction of power to idle is needed in order to bleed off the excess speed that is used during the approach phase. If the thrust is not reduced to idle, the aircraft will take longer to land, and use up more runway.
If you don't flare enough, the rate of descent will be too high and the aircraft will land hard.
Too much flare will make the aircraft climb back up.
The objective is to land smoothly, that is reducing the rate of descent which is still about 700 ft/min, 400 to 500 ft above ground. Landing with this rate is quite hard, so we need to reduce this rate of descent before touch down.
The usual way is to lift the nose, thus increasing the AOA, therefore increasing the lift but not to a point of climbing, so a speed reduction would be simultaneously required. Thus reducing the thrust is needed. Happily this is very helpful if thrust reversers are used once on the runway, since reversing requires the transition through idle.
Are you talking about a "flare"? Or a "flame-out"?
A flare is part of the landing sequence. You raise the nose of the airplane so that you touch down gently on the main gear. The engines need to be at idle (or at least, low thrust) at or before that point because the whole point of landing is to come to a stop.
A flame-out is what happens to a turbine (jet) engine when it doesn't get enough fuel to sustain combustion. In this case, the thrust is lost because the engine is no longer running, not because of anything the autothrottle did.