It's like a game of chicken. By "verge" it really means "brink".
In other words: to get as close as possible and just not quite step over the line which would cause a war.
The "running away from" is another metaphor: in this case, to back down from the challenge rather than face up to it.
If a bully gets in your face, then standing straight and not budging, however close and shouty he gets, is one thing. Brandishing a weapon and shouting back and threatening to attack is "getting to the verge". If it proceeds to an actual fight, then you've failed to "get to the verge without getting into the war".
And if when the bully shouts at you, and back down and apologise, even if it's not your place to do so, that is "try to run away from it", because you're "scared to go to the
That at least is the gist of Dulles' expressed philosophy. Whether you agree with his stance or not is a different thing altogether.