There are many variations, but the essential ingredients are that all the open brood is removed from the original hive along with one brood box and virtually all the honey so that all the field bees returning are crowded up into the supers. The nurse bees are freed up because there is no open brood. The original hive usually won't swarm because the brood nest has no open brood and there are no stores yet. The new hive won't swarm because there is no field force (they will fly back to the old hive). Some people put the queen in the new hive and let the old hive requeen itself. This is the perfect time for it to be broodless while raising a queen because it will free up nurse bees to forage. Raising bees DURING the harvest is counterproductive. Raising bees BEFORE the harvest is what gives you surplus. Some people leave the queen in the old hive and let the new hive requeen itself. Some people buy queens and requeen BOTH hives.
The cutdown part is that there is only one box for the capped brood left and no open brood, so the hive is "cut down" to just one brood box.
This technique will greatly increase your harvest, it is pretty much necessary if you want to raise Ross Rounds or Hogg Half Comb or Bee-O-Pac or any other kind of casette system. It is described in detail in Killion's Honey in the Comb and in Taylor's The Comb Honey Book and any other book on comb honey production.
Basically the old adage you can raise bees or honey is kind of true and kind of not. You can do a split and have the new one raise bees while the old one raises honey and end up with more honey and eve get more hives to boot.
The splits can be left as seperate hives, or, if you don't want more hives, they can be combined after the flow.
The most critical part of all of this is the TIMING. It needs to leave the old hive broodless (as in no OPEN brood) during the flow to free up nurse bees to forage. If you do this too early you lose a whole brood cycle of bees that could have HELPED with the harvest. If you do it too late, the bees were busy feeding, warming and caring for brood instead of foraging. Having a hive queenless and/or broodless at precisely the RIGHT time can double your harvest. Having a hive queenless and/or broodless at precisely the WRONG time can wipe out the whole crop.