Q:

How do a triggered spark gap work?

A triggered spark gap operates such that,

The secondary circuit would create an initial spark from its cathode to the anode of the main circuit. 
This will then cause the primary circuit to be able to spark the across from the primary cathode thus closing the primary circuit.

The explanation of how it works that is offered to me is that (1) will create a compression in the electromagnetic field between the primary cathode and anode thus weakening the dielectric field between the the 2 plates hence allowing the a spark to go through.
However, I am unable to visualize and draw an image to explain how that actually works, or is there another explanation on how do a triggered spark gap works?

A:

The answer has been updated based on more information and questions in the comments.
The gap between the primary (large) plates is too wide for the 30kV source to produce a spark, so some help is needed and this help comes from a relatively small spark produced by the secondary circuit. 
The gap between the sharp electrode of the secondary circuit and the right plate of the primary circuit is made small enough for the 3kV source to generate a spark. 
This trigger spark ionizes the air near the right plate and, once the air is ionized, electric field between the primary plates will quickly spread the ionization toward the left plate leading to the breakdown of the wide gap.
This will result in a sharp drop of the gap resistance and, given sufficient power of the source, an arc could be established.
To answer your questions about the E-field and about minimizing the time of ionization, we'll have to be more specific about physical implementation of the gap. 
But, first, I'd like to note that, with so high voltages and, based on your comments, high currents, this project presents many safety challenges, even if you wear a welder helmet and take care of fire and electrocution hazards. Also, my answer is based on general considerations - not on specific experience with triggered spark gaps, so it has to be taken as such.
In general, everything else the same, to speed up the ionization of the gap, it has to be made as short as possible without self-igniting at the required level of the primary voltage (30kV in your case). 
If the trigger electrode extends into the main gap, as shown on your drawing, it effectively reduces the width of the gap and therefore its voltage rating. So, I would stick with a typical physical implementation used in commercial triggered spark gaps, where the trigger electrode is more or less flush with a primary electrode, as shown on the diagram below.
 
Further, to speed up the ionization of the gap, I would make the trigger spark as big as practical, i.e. the secondary gap has to be sufficiently wide (to increase the ionized volume) and the trigger pulse has to have sufficient voltage and duration to break through it.
The diagram also shows tentative lines of the E-field. Its initial value could be roughly calculated as $E=30kV/d$, where d is the width of the main gap.
Once the breakdown occurs, or, for your application, the switch is turned on, the resistance of the gap will sharply drop and a (potentially) significant current will start flowing, the magnitude of the current being limited mostly by the load and the internal resistance of the power supply.
With that, the 30kV voltage (which will likely decrease due to the internal resistance of the power supply), will redistribute, such that most of it (or at least a significant part of it) will be applied to the load and much less voltage will remain on the spark gap. As a result, the E-field in the gap will decrease accordingly, since it will remain roughly proportional to the voltage across the gap.

