This one is quite simple, actually. For those young players at home, some transistor info first (I'll use the term "Xstr" here, as "tranny" has a different connotation here in the States).
The circuit works by having a small current flow into the base of the first xstr, where it gets amplified. The amount of this amplification is called Hfe, or Beta. For the common small-signal or general purpose types, this is typically around 100. So, the first xstr will see any stray currents - we'll guess a small value of ~1uA - and send the now 100x (100uA) to the base of the next xstr (called "stages" in amplifier lingo).
As with the first stage, the next xstr will take the 100uA and shoot 100x that (10mA) to the base of the next stage, which then allows up to 100x (1A) to flow between it's Collector-Emitter to the LED (the dropping resistor prevents the full 1A from blowing up the LED).
Essentially, what you've got here is a radio wave/EMF detector. Also, note that bi-polar transistors (BJTs) operate on current, not voltage.
Well, sort of. And this "sort of" is the whole reason you're seeing what you're seeing. You see (pun intended), in order for the xstr to start doing its thing, there needs to be a certain base bias voltage to get it to conduct (typically 0.2-0.3V for Germanium types, and 0.5-0.7V for Silicon). Without that little "kick", it just sits there twiddling it's thumbs. Although there are plenty of currents being induced in your antenna, none of them are powerful enough to kick the voltage up to 0.6V.
That is, until you pull the plunger of your (I'm guessing, here) plastic syringe. Most plastics, when rubbed, generate some huge potentials (voltages!). Your syringe just happens to generate enough kick to get that thing started. The reason it doesn't do it when pushed? An NPN xstr needs a positive voltage at the base. Pulling the plunger generates a positive voltage, pushing does the opposite! You could re-test this using PNP xstrs, and see if the behavior flips.
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* If you want to see what plastic EMF looks like, just dump some packing "peanuts" into a plastic storage bin and shake - then dump out the peanuts. The ones sticking around show you where the fields are.
** Jeez, now I'm sounding like Beakman and Jaxx!