Question

Explain why an object moves if the action-reaction forces are equal.

Answer 1

u'll find it helpful
http://www.straightdope.com/columns/read/2157/if-action-reaction-how-does-anything-move

Answer 2

Because the action and reaction forces don't act on the same object.
So, two equal and opposite forces will cancel out only if they act on the same object.
Briefly, the action force is of an object on an other, and the reaction force is of the other object on the first.