The short answer is: “Years of practice.”
The slightly longer answer is: “Physics, and years of practice.”
A bow works the same way a spring does. The moment you start pulling back on a bowstring, potential energy is stored in the flexing limbs of the bow. The instant you let go of the string, all of the stored energy is transferred instantly into the arrow, sending it flying down range. This is why it’s very dangerous to pull back and let go of a bow when with no arrow, called a “dry fire.” The energy your arm transferred into the bow has no place to escape to, and stays in the bow itself. Arrows fly pretty fast and in the video you can see how much a stiff arrow flexes after its shot, so you can tell there is a lot of energy in them. So much energy in fact if you dry fire a bow, it can shatter.
You may wonder, if your arm is the source for all this energy, how come the results are much less dramatic when you just throw the arrow? The amount of energy needed to move the string at high speeds is much less than the amount of energy needed to move your heavy arm at the same speeds. Your body can’t generate the huge amount of energy needed to move the combined mass of your arm and the arrow at speeds over ninety meters a second. But by exerting the energy over more time, storing it in the bow, and then only using the lightweight string to push the arrow, more power is transferred into the arrow.
Gravity is an important factor too, and even though Isaac Newton didn’t define it until 1687, archers have been taking it into account throughout history. The force of gravity is constantly accelerating bodies to Earth’s surface at 9.8 meters per second squared. There are no other forces acting on the arrows keeping them in the air, so even though they’re traveling very fast in a horizontal direction, there’s nothing keeping them afloat vertically and would hit at the same time as a dropped penny. Usually they just hit the target before the ground. The trick archers have found to compensate for the force of gravity is simple though, aim higher.