Minimal Coupling
Landau's version (§17). A charge couples to the field by the replacement $p_i \to p_i - \tfrac{e}{c}A_i$ in the free action; quantum-mechanically, $\partial_\mu \to \partial_\mu - \tfrac{ie}{\hbar c}A_\mu$.
Geometric identity. Minimal coupling is the Covariant Derivative of the Associated Bundle in which the matter lives: $$\nabla_\mu = \partial_\mu + \rho_*(A_\mu) \;\xrightarrow{\;U(1),\ \text{charge } n\;}\; \partial_\mu - i\tfrac{ne}{\hbar c}A_\mu.$$ "Minimal" = use only the connection, nothing else. Non-minimal couplings (e.g. Pauli term $\sigma^{\mu\nu}F_{\mu\nu}$) use the curvature as an extra ingredient.
Why it's forced, not chosen. If $\psi$ is a section of a nontrivial bundle, $\partial_\mu \psi$ is not even well-defined (compares different fibers by an arbitrary convention). The connection-corrected derivative is the only meaningful first derivative. So the coupling prescription that looks ad hoc in Landau §17 is, geometrically, the unique possibility given (bundle, connection, representation).
Gauge covariance check. Under $\psi \to e^{in\theta}\psi$, $A \to A + \tfrac{\hbar c}{e} d\theta$… the two changes cancel in $\nabla\psi$. Landau does this computation; geometry explains why it had to work.
Related: Covariant Derivative · Four-Potential · Gauge Transformation · Group Representation · Module 09 - Electromagnetism as a U(1) Gauge Theory
Sources: Landau §17; Woit (covariant derivative chapters).