I believe they are put on horses hooves to protect them from rough surfaces.

Horse shoes were made to protect the horse’s hooves, as with the cavalry, and, in some cases, increase the traction the horse could get on slippery surfaces and inclines, as with draft horses.

A horse’s hoof is actually a big toenail. As with our fingernails and toenails, rough activity causes them to split, wear down, break. For a horse, that is dreadful. Horseshoes protect them.

Once they get a fresh set, the old ones are repurposed for a game. :-)

Detailed here: https://horseyhooves.com/why-do-horses-need-shoes/

Because they didn’t want to be the only sole going barefooted.

Why not? It’s a fun game. Similar to “Washers”.

It can’t be overstated how much societies with horses relied on those horses for their way of life. A horse’s hoof is an extreme weak point, and if it is broken or infected, 99 times out of 100 it’s just been way easier to kill the horse than try and rehab it. Shoeing a horse reinforces that weak point and allows the animal to cross all kinds of terrains with huge burdens without fear of a hobbling injury. They also increase traction. Shoeing horses is probably one of the most important developments in human history.

I expect most horses would risk hoof injuries from even gentle riding if left unshod, though some horse enthusiasts do prefer it.