In quantum field theory, classical background fields are created in the following way:

Given a quantum field $\phi(x)$, the transformation that changes the field to $\phi(x)+\phi_{cl}(x)$, where $\phi_{cl}(x)$ is a classical field, is called a Bogoliubov transformation. By introducing a cutoff and a Fock space description of the field with the cutoff, one can see that the Bogoliubov transformation is a unitary transformation that transforms the vacuum state of the Fock representation into a coherent state whose expectation is the classical field. If the cutoff is removed, the transformation remains unitary if the classical field decays fast enough at spatial infinity; otherwise the transformation maps the vacuum sector to a different superselection sector.

This can be seen by working with free fields only. In this case, the Lagragians before and after the Bogoliubov transformation are both quadratic, and everything is exactly solvable.

This is also the way spontaneous symmetry breaking works - the field in an unstable (false) vacuum state undergoes spontaneous fluctuations that drive the system exponentially fast away from the vacuum state until it settles in a new (physical) vacuum state, characterized by a nonzero vacuum expectation value. This nonzero vacuum expectation value is the classical background field.