As it is seen above, even experienced electrical engineers, inlcuding myself, find it very difficult to explain physics of flow of electricity in simplified terms, because it not as simple. In fact, it is not fully understood as to what exactly happens inside the material when electicity flows, but we all know its effects and how to manouver it in limited way to our benefit.
And to answer your basic question, IRstufff had the closest analogy, the electrons do not loose their charge, but they eventually settle down with a matching proton to have a net charge of zero in the end, just like the musical chair, except there is no shortage of a chair (as no one removed a proton from the conducting path), so no electron is left out.
Explanation of electricity lies in combination of wave propagation and some particle thoery and much more. Very similar to a body of water can move energy by moving mass (ike a pumped water or a river) as well as just the waves through it e.g. a tsunami in sea) or just pass on pressure differential as in case of a hydraulic press.
In fact in the end, electricity flows when there is a temporary pressure differentical between two points of a conductive path, just like as in hydraulics. If you remove the source of pressure difference, the work stops. The nature looks for evening out the pressure difference..be it electricity or anything else.
We can go on tangent on this..forever so I will stop.