Hidden wire detector using BJT transistors
A better way to look at this is that your mains voltage  is hundreds of volts wrt ground, and only a small capacitive coupling will drive enough current through the base to illuminate the LED. There is plenty of voltage compared to the few hundred mV required to forward bias the base sufficiently.







simulate this circuit &ndash; Schematic created using CircuitLab



Here you can see that only 1pF of coupling is sufficient to allow strong current pulses through the LED. C1 represents the near-field capacitive coupling to the mains wire. C2 represents the user's body coupling to earth. 

Only a few tens of nA are required at the base, thanks to the Darlington triple transistor's high current gain.