The Art and Science of Hurling (from Physics for Scientists and Engineers with Modern Physics Chapter 3 — Applications) explains (one facet of) why a bow “throws” an arrow faster than you can throw one by hand:
The bow is probably the first mechanical device invented to achieve projectile speeds faster than those attainable by throwing. Energy supplied by the human arm is stored in the bow limbs while the bow is drawn. The stored energy is then released as the kinetic energy of the arrow. This is a very efficient process since a large fraction of the stored energy gets converted into the motion of the arrow. Compare shooting an arrow this way and throwing it by hand. In both cases its is the arm muscles that provide the energy. When you simply throw the arrow you also “throw” the arm. Since the arm is much more massive than the arrow it gets the lion share of the energy.