Looked at the video. I think I see what your asking.
The battery (or solar panels) produce DC only (direct current: does not vary in amplitude), 12 volts generally.
Remember that the spark at the spark gap device was a short but steady arc, not a single zap like the spark one sees in a spark plug. The morse code was made up of short and long spark periods
To increase a DC voltage it is necessary to, first, convert the DC to AC (alternating current: does vary in amplitude) and then (secondly) feed this AC to a "step-up" transformer. Transformers will do nothing work with a steady DC only input, except heat up.
Ordinarily, in order to generate AC from a DC source you have use some manner of self-oscillating device (in the old days, a "Chopper" 0r "vibrator" device, or nowadays a solid state circuit (555 Timer or the like) whose output is an AC signal that is , essentially, a train of on/off/on/off, etc., DC pulses. Think of a rapidly blinking light.
A "step-up" transformer has a greater number of turns of wire in the secondary (output) than in the primary (input). For instance, a secondary with 1000 turns of wire and a primary with 100 turns of wire will boost an input voltage by a factor of ten (12 volts AC in, 120 volts AC out). That 120vac could then be fed to a 1 to 100 ratio transformer (100 turn primary with a 10,000 turn secondary) to produce 12,000 vac. A "step-down" is merely the reverse.
The inverter mentioned performs the above combined functions. It takes 12vdc and "inverts" it to 120vac.
They kind of glossed over what the guy did once he had to use a battery to power the spark gap transmitter. But whatever he used, he would have had to use something similar to what I described above.