The mechanism of Kondo screening in strongly correlated molecules with several unpaired electrons on a metal surface is still under debate. Here, we provide a theoretical framework that rationalizes the emergence of Kondo screening involving several extended molecular orbitals with unpaired electrons. We introduce a~perturbative model, which provides simple rules to identify the presence of anti-ferromagnetic (AFM) spin-flip channels involving charged molecular multiplets responsible for Kondo screening. The Kondo regime is confirmed by numerical renormalization group calculations. In addition, we introduce the concept of Kondo orbitals as virtual molecular orbitals associated with Kondo screening process, which provide a direct interpretation of experimental dI/dV maps of Kondo resonances. We demonstrate that this theoretical framework can be applied to different strongly correlated open-shell molecules on metal surfaces, in good agreement with published experimental data.