Build the circuit as shown.
When you touch with your finger the soldering tips A and B the LED lights up. The skin of your finger has a resistance in the range of a few hundred kΩ to a few MOmega;. The resulting low current switches the Darlington pair through.
The LED even lights up weakly when you touch only B. Why is this? Your body acts like as an antenna, which "receives" electric and magnetic fields in your immeadiate surrounding. This induces an - albeit small - voltage.