Proud flesh is when normal tissue overgrows during healing from an injury. It generally forms in wounds on the lower legs, most often at joints. It's part of a natural mechanism to prevent infection, but because horses produce so much of this "granulation tissue" compared to others, sometimes it can produce too much...which then prevents the wound from healing. Proud flesh can be prevented by properly bandaging lower leg wounds. If proud flesh develops it can have to be surgically removed.
Proud flesh can form on other animals too, including humans. Horses are just more prone to it.