Easier than you may think!
Artificial queen rearing is simply converting a worker larva into a queen using one of a variety of methods. Some methods are described on this website and can be accessed by the button on the top left. There are far more methods than are listed here and many more variations that users have developed as they find different ways of working.
Although drone eggs are infertile, both worker and queen eggs are fertile, so identical. We are told that for the first 48 hours after hatching, worker and queen larvae are fed the same diet, therefore we can use worker larvae that are less than 5 days old from the laying of the egg to convert into queens. Many queen rearers work on the cautious side, using larvae that are around a day younger, otherwise there is a danger of producing poorly fed queens.
Bees build queen cells under three impulses - swarming, supersedure and emergency. All the queen rearer is doing is replicating these. The swarming impulse is fairly difficult to replicate as you need to get the colony into swarming mode, which is difficult to time because you have to wait until the colony responds to your actions. Some, especially the less swarmy colonies, may not oblige at all during the season. Supersedure and emergency are much easier to replicate. In general the supersedure impulse is achieved by what is referred to as "queenright cell raising", emergency as "queenless cell raising". Some methods use both emergency and supersedure impulses.
All we are doing when we rear queens artificially is to take a larva of the right age and manipulate it or the cell in which the original egg was laid and present it to a colony that we have set up to convert that larva into a queen cell.
Queens are reared artificially because there is more control for the beekeeper. You can rear more queens and with some methods you can do it in succession throughout the season. Planning is needed, together with a facility to get the resulting queens mated. This suits the commercial queen producer, the more serious beekeeper who is keen to improve their bees or a group of smaller beekeepers working together, as you may get in a local BKA.