I'm an EE with almost 30 years' experience and I just looked up the same thing a couple years ago because zig-zags are used extensively in wind farms (for now anyway, not really required anymore). I used them over the years but just assumed they worked. Take the standard WYE phasor diagram with a-phase at zero degrees and divide each phasor in two. The part attached to the origin is the positive sequence.
The second part of the phasor has the polarity reversed and is the negative sequence component. a-positive is connected to c-negative, b-positive to a-negative, and c-positive to b-negative. If you do the math you will see these symmetrical components add up to zero. That is, the positive and negative are cancelling each other out.
The zero sequence is the only component that can flow from the phase terminals to the neutral. When attached to an ungrounded system and the neutral attached to ground the zig-zag becomes the artificial grounding device. The major benefit is controlling ground fault levels.
This system with a neutral grounding resistor is used in mining for that purpose. Sizing the resistor is a discussion for another day. When implemented with the auxiliary devices, the single controlled ground point provides a level of safety, allows for selective feeder relay tripping, and makes finding the ground fault much faster. Note, fuses are not useful for ground fault protection in these schemes.
For practical purposes, the zig-zag transformer winding is a wye winding split at an intermediate point (usually the middle voltage-wise) to provide two benefits: harmonic mitigation and very specific phase control. The neutral is achieved the same way as on any wye winding but above description of the benefit is spot on.
In mining we size the NGR in conjunction with the transformer impedance to limit voltage on the frames of the equipment. Commercial sizing is a bit more complicated but there are a couple of IEEE standards about it.