Healing a cut is like repairing a part of your machinery. All animals have some ability to repair damaged parts, and this is so common that the repairing has a special name: regeneration.
In some of the simpler animals, regeneration is remarkable. A starfish can rebuild a new tentacle that is cut off, an earthworm can replace much of its body that is lost, a crab can rebuild a new claw when one is lost. In larger and more complicated animals, regeneration is more limited. You cannot grow a new arm or leg, but you do have some important repairing ability, especially for damaged skin.
Your skin is a special and important part of your body that people seldom think about. The outer layer of skin is made from the tough pieces of cells that are no longer alive. Underneath there is a layer of cells that is always multiplying and making new cells, which are pushed toward the surface. When you have a cut or break in your skin, the growing skin layer pushes new cells sidewise and these slowly close up the break.
Our bones and muscles and even some of our nerve fibers can grow more to repair themselves. Our bodies cannot repair everything, but I think it is a good thing that they can do so much. I like to think of my body as a fine piece of machinery--so good that it can even make its own minor repairs.