Electrons are not being created nor they are destroyed. The electrons already exist in the metal.
The metal consists of layers of positive ions (depicted by orange circles in the picture) in which the electrons (depicted by cyan colored circles in the picture) are in continuous motion. Hence, the charge on the metal is overall neutral.
Where did the electrons come from?
The electrons escape from the metal atom leaving the metal atoms positively charged. The delocalized electrons form a "sea" of electrons in the metal.
What is electric current?
Electric current is defined as the amount of charge moving across a cross-section of a conductor per unit time.
$$I = \frac{\Delta q}{\Delta t}$$
To have a current in a metal, you need to have a net movement of charge.
What causes electric current?
In figure one, the circuit is open. The electrons are free to move around randomly (due to thermal energy). The electrons keep bumping into atoms and keep changing directions. There is no orderly motion of electrons. Therefore, on average, there is a net zero movement of charge.
In the second figure, the circuit is closed, i.e: the metal conductor has been connected to a voltage source (electric generator or a battery). The potential difference across the conductor exerts force on the electrons which causes the electrons to move in order from a region of lower potential to a region of higher potential. Of course, the electrons still collide with the atoms but the potential difference (technically, the electric field) tries to get the electron back to the orderly motion. Therefore, there is a net charge flowing in one direction which constitutes the current.
Therefore, no charge is created or destroyed rather they are just moved. It is similar to a water in a hose. The water in a pipe does not flow unless there is a pressure/height difference across the ends of the pipe. However, the water molecules inside the pipe are in continuous motion.
An important point to note here is that the electrons move really fast (millions of meters per second) but at the same time, they collide with many atoms. Therefore, the net velocity averaged over time is very small. We call this velocity drift velocity. In copper, it is of the order of millimeters per second.
$$I = ne v_d A$$
where $n$ is the number of electrons per unit volume of the conductor, $v_d$ is the drift velocity and $A$ is the area of cross section of the conductor.
What does a generator do?
An electric generator basically creates an E.M.F (voltage/potential difference) across a conductor which produces a current.
How does a generator work?
An electric generator works on the principle of electromagnetic induction.
The law states that if the flux through a coil of wire is changing, an E.M.F (voltage) is induced in the coil.
Magnetic flux is the amount of magnetic field lines passing through a surface.
The magnitude of E.M.F induced is given by Faraday's law,
$$E.M.F =\int\vec{E}.d{\vec{l}} = \frac{d\phi_B}{dt}$$
where $\phi_B$ is the magnetic flux through the coil.
The E.M.F induced is such that the induced current produces a magnetic flux of its own which opposes the changing magnetic flux. This is known as Lenz's law.
You can go even deeper to explain why a changing magnetic flux induces an electric current. A changing magnetic flux produces an electric field.
This electric field drives the induced current.
If you are new to magnetism, refer to this answer which gives an easy way to understand magnetostatics if you are good with electrostatics
How are voltage and current related?
In simple DC circuits, they are related by Ohm's law,
$$V = IR$$
where R is the resistance of the conductor.
