Some horses are so narrow they will actually catch the inside of a front leg (generally only in front, brushing in the hind end is much, much rarer) with the opposite hoof. This is called brushing or "speedy cutting." The boots protect the horse from damage.
Brushing boots are also sometimes used on jumpers to prevent damage caused by hitting a rail, but there are also specialist boots for this. (And some stadium jumpers don't believe in putting boots on the front leg because they want the horse to feel the rail when he hits it as an encouragement to jump better next time).
The boots on the right side of the horse (viewer's left) are standard brushing boots, front and rear. Image courtesy of Maloq.