Mr Tesla up to his tricks...
It isnt any great secret how Tesla did this - it has become fairly common knowledge that a Fluorescent Tube will illuminate when in the presence of a massive electrical field without requiring any electrical connection. The bulb he's holding doesnt need one and doesnt even have one, when energised by an antenna connected to a high-voltage coil off-camera.

To generate this field allegedly requires a lot of energy. Tesla managed to do it with comparatively little by using recycling, though in an era that thought the world's resources were infinite and placed on Earth solely for our benefit it was an odd thing to do. I suppose, he knew different, another thing that put him centuries ahead of his time.
That isnt a transformer, even though effectively it does the same job using the same technology. It's a charge-pump and bears more relation to a radio transmitter than a transformer. Note that I have used a capacitor symbol connected to ground to represent the top of the circuit. This is normally a large metal ball separated from the ground plane by a few feet of air. Because it takes around 25KV an inch to create a normal arc, it requires millions of volts to puncture several feet of air as a dielectric, and a normal transformer could never achieve it without arcing out inside. The limit for these things is geometric and related to thickness of insulation and number of turns.

Because a coil carries a differential between the first and last turns, spread out over the turns, when energised neighbouring turns carry a fraction of the differential and when it gets too big it defeats the insulation between them and burns out the windings.

A Tesla Coil doesnt carry a million volt differential between the ends at any time, so it doesnt burn out. Instead of an AC source inductively powering the secondary, Tesla feeds the coil with short timed pulses that coincide with a wave of voltage that runs up and down the wire in the coil. It does this because the ground plane is capacitive, and when you hit it with a fast pulse it reflects some of it back because it cant absorb it. Voltage travels at a fixed speed in a material, and the material is very long and thin so it takes time for the pulse to reach the top and reflect back. This is how an LC or Tank circuit works; pairing a capacitor with a coil and bouncing the energy back-and-forth between them.
Obviously with a little extra energy added to a million-volt wavefront each time it passes, before long it will defeat the air dielectric and arc out. However, because it is high frequency the electrons passed into the air dont travel very far before the arc begins to decay, so effectively they arent really moving but jittering in place at high frequency. This drastically shortens the path the electrons have to take, so a much lower average voltage is needed to cross several feet compared to a DC arc, in which the electrons have to jump the entire distance.

It distributes the energy in the arc over the length of the arc the same as in the turns in the coil, a much longer distance to travel, and creates it from a very small amount of actual input.

Thats half the story, now we know that it takes voltage, and not amperage to light a fluorescent; a tube of glass coated with phosphor that reacts to electrical energy by emitting photons. It contains a vacuum to make it easy for electrons to move, and a large electrical potential is set up between two ends to move them.

Unfortunately, this is where Tesla left off. We all know the story of how Westinghouse contracted him to improve his generators on a promise he never made good on, famously ripping him off with a racist remark into the bargain. Pity the greedy fool, because Tesla only gave him half the work required to make Westinghouse an infinitely rich man. The other half was his famous coil.
Back then, power companies were more concerned with lighting homes and streets than providing current, and Tesla's bulb was a huge threat to the monetisation of that, which is also all the companies were concerned with. The evidence is still here. CFLs are designed to fail and be replaced, and that wasnt what Tesla wanted. His bulbs ran on fields, Westinghouse was trying to sell electricity, and it took over a hundred years to monetise Tesla's idea by adding heater coils that burn out like incandescent ones. All they do is further lower the voltage required to form the arc inside and use current instead to strike it. It's horribly inefficient, using 15W to strike and designed to sell more bulbs when they fail.

They dont require an arc however, and phosphor lasts forever. With a good seal, so does a vacuum, so a CFL should last hundreds of years in service as well as requiring very little energy to light. Tesla was looking at lighting homes and streets with low-cost low-maintenance bulbs as a service that competed with Westinghouse's paid system, so its no surprises who won really, but Tesla even got rid of the resistance in the wires that we still pay for today. Because his system relies on voltage wavefronts, resistance isnt an issue in cables any more than in air, so a network of synchronous Tesla coils energising the distribution cables costs very little in terms of input.

Our modern systems all have to overcome resistance to shunt power from a central generator, which is completely unnecessary, and only happens because Westinghouse was greedy and stupid. If he hadnt ripped off Tesla, we would have had a network of low power generators feeding the Grid and never had to ban Edison's nasty incandescents to save energy recently.

Back to the bodyfield experiment.

Instead of generating a huge field to excite the phosphor, you can use any old field laying around. And there are a lot of them. The human body is a huge capacitor, carrying up to kilovolts of charge in its skin and having an electrochemical factory underneath it threaded with conductors it generates an electostatic and magnetic field. It's easily big enough to light a CFL, but for the fact that it isnt moving. This is where the high voltage circuit comes in.

A radio works by throwing a wavefront up and down an antenna, which in turn makes the electrons in the air move - they are all connected by a field, so they all move in ripples spreading out from the antenna as it energises.

This is a very simple example of a radio circuit, it just energises a coil and accelerates a wavefront up a short length of cable and allows it to ground after it has reflected.

Instead of the electrons in the air, a small coil modulates the electrons in your bodyfield. This is an electrical field and therefore conformal... The coil can be as little as one turn around any part of your body; a finger, a toe, work equally well.

Note that the hot lead is insulated from everything, just coiled around a body part, and the cold lead is connected to the bulb.

You can tell which lead is which by holding the tube and brushing the terminals with the leads. The one that arcs and lights the bulb is the hot one. You DO NOT want this one connected to the bulb for this experiment, you want the cold one, and the hot one is the antenna.

I've used my wrist here to contain everything in the photo. It is also possible to touch someone, transfer the modulation into their bodyfield to light the CFL by touching it. I've managed 3 people. Can you do more?

Note that the bulb has the circuit removed and the tails of the heater wires joined together. this also works with a burned out CFL that the heater has failed in. Usually the circuit fails as it is designed to do, but sometimes you find one with blackened tubes. These still work perfectly, just not powered by AC any more.

If you have a small 1.5 foot fluorescent tube, this also works. You will need a bigger PSU than a Neon Driver to run the big ones as they have a lower vacuum in them. This is spectacular, as the tube lights only between the ground terminal and where you touch it, proving that this is an electrical field and not a magnetic one, or current that cannot flow through the glass.

The power supply amperage is unimportant, my Neon Driver produces about 4W into 5KV so its milliamps - but I've also done this with a TV Flyback transformer from a 9v battery, which is under a Watt and produces >7KV. This is enough to light a larger tube, and with a Tesla driver you only need a fraction of that...