The flow of electrons in a wire is primarily caused by the presence of an electric field. When a voltage is applied across the ends of a wire, it creates this electric field, which exerts a force on the electrons in the wire.
In a conductor like copper, the outer electrons of the atoms are relatively free to move. When a potential difference (voltage) is applied, these free electrons start to drift towards the positive side of the voltage source. This movement of electrons constitutes an electric current.
It’s important to note that the process involves a continuous supply of energy and the electrons do not travel along the wire instantaneously; they move in a series of collisions with atoms in the wire. As a result, the flow is akin to a wave moving through a crowd where each person (electron) moves only a short distance before bumping into someone else.