The issue is that in order to be useful as a grounding transformer, the design of the transformer must 'close the circuit' for any ground fault current, in other words somehow connecting a fault from one phase to the other phases, while at the same time maintaining the desired phase to neutral voltage. Another way to think about this: any power delivered by a phase to neutral fault must ultimately be supplied by a phase to phase power flow, and the grounding transformer must somehow convert this phase to phase power into phase to neutral power.
Start with how a transformer functions. With an alternating voltage applied to a coil, a _small_ alternating current flows, which produces an alternating magnetic field. This alternating magnetic field generates a voltage in the coil, and this generated voltage acts to oppose the voltage applied to the coil in the first place. This self induced voltage almost exactly equals the applied voltage, with the very small difference pushing the small magnetizing current through the primary resistance. The magnetizing current will always naturally adjust so that the self induced voltage is in balance with the applied voltage.
Any other coils wrapped on the same core will have induced voltage as well. If any current flows on one of these other coils, then that current flow will change the magnetic field in the core. If you connect a load to one of the 'secondary' coils, then this current flow will tend to decrease the magnetic field in the core. The lower magnetic field strength means less voltage induced in the primary, so more current flows in the primary to restore the magnetic field strength. The net result is that the magnetic field strength remains virtually constant, with current flow in the primary almost exactly balancing the current flow in the secondary, adjusted for turns ratio and small losses.
To summarize: a) The voltage applied to a coil in a transformer must be matched by the voltage that the transformer magnetic field induces in that coil and b) for any sort of significant current to flow on a coil in a transformer, some balancing current must flow on another coil wrapped on the same core.
I like to think about zig-zag transformers by first considering what would happen if you tried to use a simply 'wye' to derive the neutral.
A simple 'wye' transformer (this doesn't work). Take an ordinary 3 phase wye, connect the phase terminals to your ungrounded delta, and use the neutral as your ground. During normal operation, magnetizing current will flow through all three coils, and since the design is balanced, the derived neutral will be pretty well exactly between all three phase voltages. This derived neutral would be very similar to one derived using a wye connected set of resistors; just built with inductors.
Now consider a fault. We need significant current to flow through the neutral, which means through the transformer primary coils. But this cannot happen without some other coil on the same core, carrying the 'balance' current so that the net magnetic flux stays almost constant. A simple wye won't properly carry fault current.
Now consider the zig-zag transformer. Each transformer core has two connected coils from different phases. The same rules apply as for other transformers; the voltage induced in each coil must produce a voltage that nearly balances the voltage applied to that coil, and for significant current to flow in a coil you must have the corresponding 'balance' current in another coil.
In the zig-zag transformer, current in one phase's coils will be properly balanced by current flowing in the other phases. Going to your diagram, say we have some current on phase A. The phase A current on transformer 1 will be balanced by phase C current, the phase A current on transformer 2 will be balanced by phase B current, and the resulting B and C current will balanced on transformer C. The net result is that this neutral current can flow without disrupting the magnetic flux developed in the transformer cores.
So the zig-zag transformer combines developing the proper neutral voltage at no load with the ability to carry significant neutral current.
-Jon
