The explanation from F*ckYeahFluidDynamics:
When an arrow is fired from a bow, as in the high speed video above, the sudden impetus of force from the bowstring causes the arrow to flex and vibrate as it is fired. The aerodynamic forces generated by the fletches straighten the arrow’s flight, helping it reach the intended target accurately.
See the Archery Report for more information on the physics behind the bow and arrow.