Nuclear Pore Complexes (NPCs) enable rapid, selective, and robust nucleocytoplasmic transport. To explain how transport emerges from the system components and their interactions, we used experimental data and theoretical information to construct an integrative Brownian dynamics model of transport through an NPC, coupled to a kinetic model of transport in the cell. The model recapitulates key aspects of transport for a wide range of molecular cargos, including pre-ribosomes and viral capsids. It quantifies how flexible phenylalanine-glycine (FG) repeat proteins raise an entropy barrier to passive diffusion and how this barrier is selectively lowered in facilitated diffusion by the many transient interactions of nuclear transport receptors with the FG repeats. Selective transport is enhanced by "fuzzy" multivalent interactions, redundant FG repeats, coupling to the energy-dependent RanGTP concentration gradient, and exponential dependence of transport kinetics on the transport barrier. Our model will facilitate rational modulation of the NPC and its artificial mimics.