I presume that in real life the rider tells the horse which cow to isolate and where to take it. I am presuming this is done to take cows away from the herd for a practical reason like vet care, not for the fun of it.
I will see what
says, but I think this originated in having to brand the cattle, and that's why they are often (or always?) working calves. So, once a year you'd bring all your new cattle into the corral, then you'd have to get them separated one by one so you could brand them (and, yes, presumably give them any vet work they needed).
What I wonder is, these horses that do this work in an arena, do any of them actually do real work on ranches as well, or is the skill only for show purposes?
Correct, it originated in sorting cattle for any reason you would need to pick specific cattle out of a herd and do something with them like load them on a truck for example. In the branding pen calves are usually roped because if you have good ropers it is much faster.
Cutting as in the sport is basically just that meaning if they are a Cutting horse they pretty much do it in an arena. Like Reining and Reined Cow Horse the big money is in the 3 and 4 year olds. At the end of the day, what drives these sports is winning the futurity. Winning it basically turns that horse into a profitable breeding business for the owners, especially true of stallions. For example the horse I linked is named Metallic Cat and his stud fee is $15,000.
Unfortunately that also makes these sports prohibitively expensive for average people at the competitive level.
There is a market for "cutters who don't make the cut", and some of hoses do get used in ranching. They are naturally gifted at working cattle and popular with people who need a young working horse that is already trained. I would still rather have a seasoned horse off one of the better ranches like the Wagoner personally though, but that is off topic.
As for what they are doing, the rider's most important job is to know cattle well enough to pick out a cow that is going to be a good match for the horse. There are also rules about having to go into the herd to cut them, you can't just pick ones off the outside. You don't want a boring dud, or one that is going to do crazy stuff, neither will give much of a score. There is basically no score for the person, it is all how well the horse works and the horse should work the cow as much as possible by themselves once the cow has been cut out of the herd.
They change out the cattle after so many runs to give competitors a fair chance. Each time they do someone rides through and around them to settle them down and get them used to a horse being near them. Cattle get dull or sour to being worked fairly quickly so they have to do this and it's another reason Cutting is expensive.
It would be a much different sport if the cattle were picked for you and the rider's job was to take them some place. In that regard events like Team Penning and Camp Drafting are more realistic to ranch work in my opinion.