Horses have the largest eye of any land animal and their eyes protrude slightly. While there aren't a lot of one eyed horses, horses are more likely to lose an eye to injury than, say, humans or dogs. (This is likely to be particularly true when horses are used in warfare, and war horses wear head armor that is designed in part to protect their eyes.
Eye loss is particularly common in horses that have lost vision in that eye due to an illness. Without the blink reflex to protect the eye, horses are particularly prone to corneal damage which can then result in infection and the need to remove the eye. Some vets recommend removal of an eye that is totally blind before damage occurs because it is just that common and is more humane for the horse than waiting for damage.
It's common, but not absolutely required, to replace the damaged eye with a prosthetic. If a prosthetic is not applied, the sunken eye socket can look quite unnatural, but this is purely a cosmetic concern.
Most horses adapt well to only having one eye, although they may lose the ability to assess heights well enough to compete over fences. They can be spooky to start with and handlers should always keep in mind the loss of vision. (Horses with only one eye will start easily if somebody approaches and makes a noise from their blind side, although I know one "cyclops" horse who has maintained his record of only spooking once in the last 15 years - but that's an exceptional horse to start with). Horses that had partial vision often become less spooky after the eye is removed.
Eye removal is often performed using only sedation and local anesthetic (general anesthesia in horses is dangerous).
It is not in any way cruel to remove a horse's eye if it is damaged (and can be life saving). It is also not cruel or abusive to continue to ride and work a one-eyed horse normally as long as the handler remembers not to startle the poor guy from his blind side.