When the teams on each end of the rope exert exactly the same force ... in opposite directions ... the net force on the rope is zero, and it doesn't accelerate in either direction.
Equilibrium refers to a state of balance where there is no net force. The forces acting at point in equilibrium are opposite and equal.
Therefore, for a body at equilibrium the vector sum of all the forces acting on that body must be zero and also the vector sum of torques on the body must be zero.
In a tug of war for example, the equilibrium will be achieved if the two teams involved on either side apply equal forces in opposite direction, such that there will be no net force on the rope, that is the net force on the rope is zero.
