The withers are the little bump you see at the base of the neck on a horse. What they actually are is a series of bony spines that extend upwards from the vertebrae in that area.
Withers are unique to domesticated horses.
As you can see, this zebra's back is completely flat - no "bump." (Source: D. Gordon E. Robertson via Wikimedia Commons).
Same with this donkey. Both animals have a completely flat back. (Source: Adrian Pingstone via Wikimedia Commons).
And again, Przewalski's horses - wild horses - no withers. (Source: Ancalagon via Wikimedia Commons).
So, the wither is something we've bred in to our horses. Why? Is it to hold the saddle in place? Not...entirely.
This Norwegian Fjord mare has a dip in her back for the saddle, but only the slightest hint of withers.
This is a foundation Quarter Horse, with distinct withers.
And this is an Akhal-Teke, representing a breed that's the ancestor of the Thoroughbreds, and clearly showing the "shark fin" wither classic of racehorses. (Source: Artur Baboev via Wikimedia Commons).
Draft horses and ponies have very low withers. Racehorses have very high ones, which should clue you in. The spines actually anchor the top of the shoulder muscles. This increases the reach of the front legs, which increases the length of the stride, which...makes the horse faster.
So, no, they aren't to hold the saddle on - in fact shark fin withers make saddle fitting just as tough as what we call "mutton" withers (when the horse has little or no wither height).