First of all when a man push the box, the box exerts an equal but opposite reaction on the person, the action and reaction. If this was the only force in action the box would indeed remain stationary, however there is another force between the man and the ground. The person's leg press down on the ground and the ground pushes back on the man.This force prevents the man from moving.
The box will move forward when the force exerted on it by the man is greater than the frictional force between the box and the ground. Remember, there are two sets of forces acting on the box as well: the action of the man on the box and reaction of box on man, and the action of the box on the ground and the reaction of frictional force of the ground on the box
We can conclude as while the box is pushing against you it is also pushing against the surface it is on and the surface is pushing back. Therefore even though pairs of action-reactions cancel out, the net or overall force resulting from different interactions allow movement.
Here the action-reaction pair deal with two different bodies. The forces do not cancel out. We should only be summing up forces on a particular body. So, the box experiences a force from me, and if that force overcomes friction, the box will move!