The terminology is confusing because outside of the realm of component to component interconnection, the term "balanced" is meaningless to the point of misleading.
What they're referring to is differential signalling.
Generally headphones are single-ended; you have a signal wire and a ground wire. The signal is AC, and generally swings both positive and negative of the ground reference.
"Balanced" is what they say when they mean that the signal is differential -- Rather than a signal referenced to ground, both wires carry half the signal, with what was formerly the ground wire now carrying a negative mirror of the positive signal.
So, you have twice the amplifier components, twice the gain, and twice the current capability, at a cost of only 2x what a single-ended setup costs.
This is actually how many - possibly even most speaker amps are designed. This configuration used to be called push-pull.
Since there isn't a single ground reference anymore, the three-conductor connector has to go. You could use a single four-pin connector (and in fact they used to use a single 5 pin connector for both single-ended and differential driving of headphones in europe), but in a stylish holdover from the studio signalling history of this type of connection, people generally use two full-size three-pin XLR connectors.
There is additional confusion with regard to the way sometimes the amp is really just dual mono, with two separate single-ended signals. This isn't nearly the same thing, even though you hook it up the same way.
In component-to-component interconnection, 'balanced' signalling has the advantage of 6db better noise rejection. Headphones can't take advantage of this.