The main reason for doble cutting is greenspeed. As mentioned above the amount of growth on a green cut at 0.125" vs say a lawn at 3" is more noticable at the lower height. In general terms, the less grass, the smoother the surface and the faster the green.
The main thing you need to consider are the mowers. Most likely a home lawn is cut with a rotary machine with esentially whacks the grass like a sythe which isn't very precise. A greens mower is a reel mower so it cuts like a sissors and can cut as low as 0.06" (although no one goes that low) making it a very precise machine, therefore small increments have can make a big difference.
So why double cut? The clip rate of a reel mower dictates the quality of the cut; this is how frequently a blade cuts the grass. There are a lot of factors at play here including the amount of blades, the speed of the reel, the speed of the machine. Due to this, even a very high clip rate isn't perfect as there will always be a (miniscule) gap between each blade coming around to clip the next piece of turf. This is why double cutting or more is performed at times to increase speed as it cleans up these little gaps or peaks in the cut producing a better cut. The modern hybrid mowers can change the clip rate as the reels are electric and can be sped up independantly of the forward motion, which can negate some of the need to double cut. So if say a green grew to 0.13" overnight and they were mowed at 0.125" the next morning - 0.005" was removed. Now if you immediately went and cut that green again at 0.125" there would clippings in the basket. The amount would be a lot less than that first cut but it mower would still cut a quantifiable amount of grass as it cleans up those gaps or peaks from the first cut. Each cut after that would still get grass with the amount getting smaller each time as the gaps/peaks get mowed off. Usually double cutting at one time is enough as after that the gains are usually minimal. Double cutting twice a day would yield results as it cleans off the growth each time.
There are other factors too like lowering height of cut, the amount of turf a reel mower can cut at one time (rule of thumb is less than 1/3 of it's current height). Also groomers (little verticutting blades in front of the reel) or brushes can stand the grass up to produce a cleaner cut but they are still limited by the gaps in the clip rate.
Lastly it should be noted that this all adds extra stress to the turf so it is ok for special events but it is not practical on a daily basis due to cost and the stresses added to the turf.
This is a very simplified version of what is going on but hopefully it helps. This is a great resource if anyone wants to know all the details behind how greens mowers work.
https://cdn2.toro.com/en/-/media/Files/Toro/Commercial/education-technical-references/service-training-guides/09168sl_5-4-2018.ashx?la=en&hash=1531E3CD624D41D0E5C07157493AFECD0FF67BEE