Transistor with emitter attached to another transistor's base
I don't see a bias network on the base of the driver 3904, so your finger, and the resistance of your body, is giving the base of the 1st  3904 the .7 volts it needs to saturate and turn on the junction, allowing current to flow through the collector-emitter junction and driving the 2nd 3904 on, doing the same, passing current through the diode and lighting it up.