For want of a nail, the shoe was lost.
For want of a shoe, the horse was lost.
For want of a horse, the man was lost.
For want of a man, the battle was lost.
For want of a battle the kingdom was lost
All for the want of a horse-shoe nail.
(Several variants exist, this is the one I grew up with).
So, why is a horse lost "for want of a shoe." To understand, put one shoe on and walk across the room. Not very comfortable, is it.
A horse that loses or "throws" a shoe has one foot slightly higher than the other (horse shoes are about half an inch thick). This causes discomfort for the animal, especially if working on harder surfaces. Some horses come up outright lame right away. Continuing to work a horse with a thrown shoe can cause soft tissue injuries. In some cases, you can pull (remove) the opposite shoe to solve the problem, but much of the time, you're stuck with a horse you can't work until the shoe is replaced.
If that happened in the middle of a battle, then the cavalryman or knight would, at best, have to go to the back of the lines for a remount, or continue to fight on foot - which could well turn the tide of combat.
(If you're writing, this is a good way to put a horse out of action for a short period of time without actually hurting it).