What is a "flare-out" and why does auto-thrust reduce thrust during flare-out?
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.