Why do Buddhists believe it is so important to follow the path of nonviolence?

Answer 1

There are concepts that must be discussed: Force, violence, non-violence an passivity.

• Force and Violence: There are situations where violence may be necessary.It is necessary to distinguish force from violence. A prime concept in there idea of violence it contains a bad intent (anger or hatred). Once this consideration is internalized and eliminated force is separated violence.
  
• Nonviolence and Passivity: Some people react to situations requiring force with passivity, thus allowing bad situations to continue. Buddhism is against allowing harm to occur and so passivity may be more evil than the force required to reduce or eliminate the initiating event However violence may in many cases be substituted by reason, intervention and other remedies. This eliminates the need of force (or violence) to overcome harm

As a consequence Buddhists believe in reducing harm by non violent means. This may require the application of force (restraining a violent person) or rational processes (talking the same violent person into acclaimer state). It does not mean to be passive to the harm to oneself or another.