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.