I don't understand how is generated the high voltage. As far as I know, transformers only work with AC voltages. The box uses 4 AA Batteries of 1.5 V each one, so I have 6V DC powering the circuit, I think this is where the TIP41CG power transistor comes, but I don't understand how to interpret it. Also, the manual only says "Transformer", but not what kind of transformer is .. How do I know that?
The transistor and the transformer are a blocking oscillator.
That is the source of the AC needed to generate high voltage through the transformer.
The signal it generates is a series of pulses. It isn't a nice, smooth signal like a sine wave. The transformer doesn't (much) care. It works with any varying voltage.
The transformer has to have a high ratio of turns (many times more turns on the secondary than on the primary.)
I wouldn't try to duplicate the circuit with just any random parts. It most likely wouldn't work, but if it did then you can't be certain that it won't deliver a lethal shock.
With the kit, you can sort of expect someone to have made an effort to ensure that it is reasonably safe.
If you use whatever transformer you have in the junk box then you have no guarantees at all.
I don't know enough about it to tell you how to design one to be safe.
Lets redraw your circuit as shown, simply to emphasize the DOTS often used in understanding transformers. Dots indicate "the polarity is the same", and you use only one dot per winding.
If we think of the primary (left side of transformer) as built of two winding that happen to have a center-tap, those two windings having been wound with a continuous direction around the core, then we are guaranteed to have the DOTS as I drew them.
The DOTS guarantee the base will be given MORE voltage, when the collector moves toward Ground.
Since the base is DC_biased at +6 volts, the transistor is ON. As the current builds up, the collector voltage drops toward zero, and the base voltage rises above +6 volts toward +12 volts, increasing the base current and also increasing the collector current.
What will turn this OFF? to cause the "blocking"?