I imagine there are many reasons why people do serpentines with their horses. I'll share some of the reasons that I may do them.
I may be working on a horse's flexibility or on how supple they are. If a horse is a bit one-sided, turning more nicely to one direction than to the other, I may do serpentines to either a) help diagnose the problem, or b) improve the turns to the stiffer direction.
I use serpentines, at times, to build up a horse's hindquarters. Making it reach under him/herself to push through turns more efficiently.
It might be that a horse is just out of shape and serpentines are great at building up the smaller muscles that go into being really well-balanced, giving the horse a better chance at becoming athletic.
Perhaps, the horse just doesn't neck rein well. Serpentines can be used to sharpen those skills.
They are good for arena sour horses, too. Changes things up, and depending on the direction you go, it's a good exercise. Keeps the horse focused on the rider and not on the gate or whatever it has issues with.
Serpentines are just like many other 'exercises' and 'patterns' that help trainers get horses very 'bridle-wise' and responsive to reins and legs.
I use them to advance 'leg-yielding' exercises. Doing serpentines lets me put 'shape' into a horse before it makes the loop at the change of direction. It helps me develop nice 'round' direction changes that do not have any 'square' corners.
On very advanced horses, I like to lope a serpentine without letting the horse change leads. Like a counter canter on a circle, it keeps horses from anticipating lead changes and direction changes.
Serpentines are very useful exercises, but like other exercises, they must be done correctly. This means that the horse must be 'straight' when he is supposed to be straight and must have the proper amount of 'bend' and softness where he is making the 'loop' at each direction change. They are very useful at teaching a horse to stay between the rider's reins and between the rider's legs. They require a rider to be very 'self-critical' if they do not have good over-sight from a second person.
This exercise works out on the trail, also. You have to have worked with this at home so the horse knows what is being asked, but if you are out riding and the horse maybe is showing signs of reverting to some previous unwanted behavior, asking for these changes of direction helps to bring their attention back to what you are asking and not their own agenda.
I've found them extremely useful on the road, which is where I ride 99% of the time. Like the trail, scary things can pop out and doing a familiar exercise helps to get back to 'work' mode.
Also, it's very easy to add-on movements to the serpentine. For example, at each bend apex, ask for one leg yield or shoulder in, and then continue on your path. Eventually, you can ask for many leg yields (or whatever) at each interval
I do it because it incorporates both bend and straightness in the one movement.
You can do as many loops as you want, and the more loops you do, the bendier and more advanced the horse has to be. It's really hard to get that perfect serpentine, but if you keep working at it, you'll get it in time. Have fun doing your serpentine's!
It's one of those exercises that you can work on a lot at once without feeling like you are.
I sort of use them to work on developing straightness. I trot out on a nice loose rein and give my horse a change to wander. If my horse veers left I steer her right. If she veers right I steer off to the left. I don't just correct to the line i'm going I go way off the opposite direction. In this, serpentines aren't the exercise but it often looks that way on a green horse. I do this about every day with every horse. The more you do it, the more your horse will stay straight without any correction from you.
Of course it helps with steering on those greenies, I do big loopy serpenties on them at first before incorperating sharper turns to keep them balanced.
On a high headed booger that hollows their back and isn't inclines to track up, it really helps. It'll make them step up under themselves and use themselves better. As a result the head will come down and you'll get a lot accomplished without any fight. It also helps a horse that's stiff through the turns and doesn't want to put a bend in their body.
If I have a horse that's hot and is always wanting to charge ahead i'll do serpentines. If they're being speedy i'll do smaller turns. This will make the horse rate themselves down to keep themselves balanced. When they're being good on a loose rein i'll do larger sweeping turns.
It balances up your horse. If i've got one that stiff on one side instead of doing traditional serpentine i'll circle the bad way till I get my horse getting soft then off we'll go again. I'll do a turn to the good side, circle the bad etc till the horse i s moving a bit better then do 50/ 50 on each side.