We develop a microscopic theory of spin-torque ferromagnetic resonance (ST-FMR) in nonmagnetic-metal/ferromagnetic-metal heterostructures. Tracing out the conduction-electron degrees of freedom in the FM, we derive an effective interfacial exchange coupling between the localized spins and the conduction electron spins in the adjacent electron system. Based on this interaction, we calculate the current-induced driving torque, resonance-frequency shift, damping modulation, and resulting dc voltage. As a concrete example, we apply the formulation to a disordered Rashba two-dimensional electron gas and demonstrate that the ST-FMR spectrum reflects the dynamical spin responses of the adjacent electron system. Our formulation applies to a broad class of heterostructures and establishes a unified microscopic framework connecting ST-FMR spectra directly to the electronic spin responses of adjacent systems.