Dear Student,
When the bow is drawn it sores potential energy.When the string is released it converts the potential energy into kinetic energy which direct the arrow to it's target.
More accurately, Elastic potential energy is stored in the bow as a result of drawing the bow .When we draw the bow it gets deformed and the energy is stored until the force is removed and the bow springs back to its original shape.This happens when we release the string. Now the potential energy is converted into kinetic energy which moves the arrow.
Regards.