I'm not sure if you're asking what affects the breakdown voltage of transformer windings, or why its arcing at all?
In terms of insulation you have the insulation of the wires - usually enamel, sometimes teflon for when a very high voltage is generated, and layers of insulation tape between winding layers. If you have say, 500 turns on the secondary, with 50 turns on each layer, and you're developing 1kV with it (depends on how you're driving it, as a coupled inductor, or transformer). That's 2V per turn. That means the insulation between windings only has to cope with 2V. But a layer is 50 turns, at 2V each thats 100 volts for that layer. As you wind to one end, then back over it, you now have two layers with 2x100V = 200V between them at that end. Perhaps the enamel insulation can cope with that so you would insulate every two layers.
Another method is the split bobbin:
http://modelengineeringwebsite.com/Blokker_I_C_ignition_files/origineel_blokker_circuit.jpgIt breaks up the coil into sections, with that one 6kV, each section is 1kV from the inner most winding, to the outer.
To further prevent internal breakdown, you can immerse the whole coil in oil, or pot it using a vaccuum pump to remove all the air (because air is easier to punch though that the potting compound).
If you're asking why its generating such a high voltage..
With nothing connected to the secondary you essentially have an inductor, and driving it with a square wave via a switch (be it MOSFET, bipolar etc..) you're building up the magnetic field, then when the switch opens it collapses generating current in both coils - that has to go somewhere, so the voltage rises to whatever it needs to be to get current flowing. This could be an arc at the secondary terminals, or an arc across the switch (breakdown of the MOSFET). You can get several kV with an input of 12V with a turns ratio of only 1:10 - if you have the secondary connected to a spark gap that is the correct distance apart so it breaks down at a few kV. But at the same time, the primary's voltage rises too, say if you somehow manage to get 1kV from a 1:10 transformer, the primary voltage would shoot up to 1kv/10 = 100V - that can kill your switch.
Driving with an AC waveform, push pull, you're building up the magnetic field, and then 'draining it' back out, with the output secondary voltage being a product of the input voltage across the primary, and the turns ratio.
Driving a transformer as a boost converter with the secondary not connected to anything and no voltage clamp across the switch, it'll either destroy the switch, or.. it will arc internally between the layers, burning away insulation, and making further arcing easy, or rather, reducing the voltage it needs to jump.