The recent discovery of high-${T}_{c}$ superconductivity in bilayer nickelate ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ under high pressure has stimulated great interest concerning its pairing mechanism. We argue that the weak coupling model from the almost fully filled ${d}_{{z}^{2}}$ bonding band cannot give rise to its high ${T}_{c}$, and thus propose a strong coupling model based on local interlayer spin singlets of Ni-${d}_{{z}^{2}}$ electrons due to their strong on-site Coulomb repulsion. This leads to a minimal effective model that contains local pairing of ${d}_{{z}^{2}}$ electrons and a considerable hybridization with near quarter-filled itinerant ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ electrons on nearest-neighbor sites. Their strong coupling provides a unique two-component scenario to achieve high-${T}_{c}$ superconductivity. Our theory highlights the importance of the bilayer structure of superconducting ${\mathrm{La}}_{3}{\mathrm{Ni}}_{2}{\mathrm{O}}_{7}$ and points out a potential route for the exploration of more high-${T}_{c}$ superconductors.