Gregsan's and Kieran's answers are insightful analogies and the pushy electrons are certainly part of the answer.
There is another aspect to the "decision" process and that is the propagation of electromagnetic waves. There is a chapter in the second volume of the Feynman Lectures on Physics - I don't have it with me but the relevant section will be just after the Poynting theorem and energy flux is introduced - where Feynman discusses a simple DC circuit and what happens in detail when the switch is closed. This would be a good reference for you.
In a very real sense, energy is NOT being transferred through the wires, it is travelling through free space around the wires: the electrons in the wires feel the propagating field and are shifted accordingly, thus setting up the usual high conductivity boundary conditions which then guide the "probing" waves. Every change in a circuit begets waves propagating at the speed of light, which then "probe" the rest of the circuit and charge thus may or may not shift in response to the wave. So when you touch the ground, a wave propagates back towards the power source, and a complex sequence of scattered waves bouncing back and forth between you and other parts of the circuit as the new, lethal, steady state conditions take shape.
Transmission line theory is a good starting point to thinking along these lines by giving you simple wave solutions which let you thing about changes "telling" the rest of the circuit through a "notification" message propagated as a voltage/current wave which, together with the (theoretically infinite) sequence of scattered waves it begets, leads to the shift in steady state conditions.