it doesnt always.
ive cut myself on paper and it hasnt been deep enough to make me bleed.
its just separating the skin from the blood.
i tell you what though some of those paper cuts dont half come keen :|
Blood is pumped through the body in capillaries, veins and arteries. When you get cut, you are breaking one of these vessels (like if a pipe has a crack and water comes out) and so the blood comes out.
The blood is flowing through your body and when cut it foes there to clot the spot up so it can heal.
The body is literally riddled with blood vessels ... when you sever these, blood leaks out.