I've got a big essay on the topic about half done. Here are the quickie answers:
You get harmonic currents whenever you have a 'non-linear' load, one in which the current flow is not exactly proportional to the applied voltage. The applied voltage might be a nice perfect sinusoid, but the current flow is some ugly mess of spikes and square waves. This ugly waveform can be considered as a sum of many nice clean sine functions, each of which has its own frequency and phase angle.
The wye transformer secondary applies is output voltage to the various loads, and the current that happens to flow is how the various loads respond to that applied voltage. The transformer secondary might produce a perfect 60Hz sinusoidal output voltage, but the loads can still simply draw a non-sinusoidal current in response to this output voltage.
Transformers are almost perfectly bi-directional. The primary is magnetically coupled to the secondary, so voltages applied on the secondary side will certainly induce voltages on the primary side. This can go in both directions at once: I can feed 60 Hz on the primary side, inducing 60Hz voltage on the secondary side. The 60Hz on the secondary side causes current to flow in the loads, and if the current flowing in the load is not sinusoidal, then the harmonic currents on the secondary side can induce harmonic voltages on the primary side.
The third harmonic on the secondary side that is considered a problem is the 'zero sequence' portion. This is the portion of the third harmonic for which the phase angle difference between all the phases is _zero_ degrees. If the loads are all perfectly balanced, then _all_ of the load third harmonic will be zero sequence. The zero sequence harmonics place the _same_ current flow on each terminal at exactly the same time. Since all the terminals are 'in phase', rather than nicely balanced 120 degrees out of phase, this current flow does not balance out. Instead it simply shows up flowing in the neutral. Third harmonic loads the neutral in the same way that the neutral would be overloaded in a multi-wire circuit with all of the hots fed from the same phase leg.
Third harmonic currents cannot _enter_ a delta connection. This is because there are only three terminals on the delta connection, and everything is in phase for the third harmonic, meaning no phase difference. With no phase difference there is no voltage difference, and thus nothing to push the current flow into the delta.
But this also means that any third harmonic current already flowing in the delta connection cannot couple to the outside world; the third harmonic current flow does not produce any third harmonic voltage at the terminals of the delta.
Third harmonic current flow in the wye secondary will induce third harmonic current flow in the delta primary. But the phase angles are such that the current flowing out of the head of the phase A coil is exactly balanced by the current flowing into the tail of the phase B coil. The third harmonic current flows in the coil set, but doesn't produce a voltage difference that effects the current flow into our out of the primary terminals. This is how the third harmonic gets 'trapped' in the delta primary.