Power is applied to the Tesla coil system by the primary transformer. This voltage, although high, is not nearly enough to jump the spark gap so that it can get to the primary coil. Instead, the only place that it can go is to the capacitors, where it builds up until it is able to jump the gap and run through the primary coil. When the power runs through the primary coil, it sets up a magnetic field inside the coil.
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But, since the spark has to go out, this field soon collapses. But the secondary coil is positioned to catch this EM wave as it moves inwards. The secondary coil is designed to convert the EM energy back into electricity, which causes current to flow though the second circuit and create the HV output.
Repeat process a thousand times a second.
The reason that the output of the secondary is greater than the output of the primary circuit is because the secondary coil has so many more turns than the primary coil. If the primary coil has 10 turns, and the secondary coil has 1000 turns (which is 100x as many), then the voltage flowing in the primary circuit will be multiplied by 100x when it reaches the secondary circuit.
Be careful with the websites, as some tend to have bad information. Although you are right about the output of a Tesla coil being high voltage but a safe current, the primary circuit is a very high current and must be avoided while the coil is on.
There is this tutorial that contains step-by-step instructions on how to make a Tesla Generator. The guide is called: Tesla Secret Handbook.
Since the publication of The Tesla Secret Handbook, tens of thousands of people allover the world have used it to build their own personal Tesla Generators and are now powering their homes for free!
Do you want to find out how you too can generate free electricity to power your house at no cost? If yes, then you need to get a copy of the Tesla Secret Handbook.
Click here: Nikola Tesla Secrets, to read more about this book.