The electrons in an atom are loosely bound to the nucleus. They revolve around nucleus. They have some energy. In metals these electrons are loosely bound. They are easily excitable when they get some more energy. When electric potential is applied across a conductor wire, due to the electric field and electric force they jump from one atom to another. They travel at the drift velocity.
At any point of time they are in some atom of the same material, far away from the nucleus, and they are shifting.
When you cut the wire, the potential difference is zero across conductor, there is no electric field. So there is no more electric force on electrons. So, in which ever atom they are when wire is cut off, they stay there. To jump to next atom and move they need more energy. There is no extra energy when power is off.
The electrons do not fall off because the attraction power between nucleus and electrons in the atom of a conductor is more than the attraction force of earth on electron. It is also more than attraction force between air molecules nearby and electron in the conductor.