When Morgenthau is referring to the “holder” of the balance, he is talking about one state that exists in a multipolar system. It exists to assure that the balance of power is constantly maintained, regardless of one individual state’s current level of power. The holder achieves this by continually shifting alliances, making sure that they are always aligned with the state(s) that would be at a disadvantage without them. Then, if that faction begins to tip the balance of power too far in their favour, the “Holder” of said balance would be obligated to break its alliance and join the other side for the good of all the principle actors. This way both sides of the balance maintain an equilibrium that is moderated by the “holder”.
The most common and widely used example of this comes from examining the 19th century balance of power system. During this period, Britain is generally acknowledge to be the holder of the balance, as it shifted its alliances frequently in order to decrease the likelihood of war and to make sure that no single state rose too far above its counterparts. Many IR theorists believe this is one of the main reasons that particular era (before the outset of WWI of course) was relatively peaceful.
This is a very generalized and brief description of what being the holder of the balance entails, much more could be said on the topic. Here is an example of a more in-depth analysis.
