I can answer the “why”. Someone else will have to go into more detail on the “when”, but I can do the why.

Your average horse...he walks on his toes. Or at least, on one very large toe. The hoof is the nail of that toe, surrounding and protecting the sole of the foot and a soft leathery-fleshy bit called the “frog”.

Here’s a diagram.

Just like your toenails, the hooves of a horse grow continuously. If they get too long, they can interfere with the horse’s gait pattern. If they break off too short, like when you break a nail, there’s nothing to protect the soft, sensitive parts of the foot - a cut or bruising/inflammation on those parts will cause a LOT of pain, and can make the horse all but unable to take weight on that leg at all.

A wild horse keeps its hooves in check fairly naturally. They wear down as it runs.

A domesticated horse is likely to spend much more time on hard paving or roads than a wild horse, and is more likely to carry the heavy burden of a rider, a cart or other drawn vehicle, or a pack-saddle. Both of these factors put more strain on the feet, and wear the hooves down faster. Horse shoes work by putting a metal “cap” over the hoof; this cap protects the feet and keep the hoof from being worn away so quickly that it hurts the horse. Instead of the hoof taking the weight of the burden, or contacting the hard road...the shoe does, and the shoe won’t wear away.

A farrier (specialist shoeing blacksmith) will take a look at a horse’s feet about every six weeks. My family’s horses are seen that regularly. He’ll take the shoes off, trim the hoof back evenly (with the shoe on, the hoof keeps growing!) inspect the health of the horse’s legs and feet in general and nail the shoe back on. The nail doesn’t hurt, as long as the farrier is competent and only goes through the hard layer rather than the squishy frog.