A microscopic theory for the NMR anomalies of the planar Cu and O sites in superconducting La1.85Sr0.15CuO4 is presented that quantitatively explains the observations without the need to invoke antiferromagnetic spin fluctuations on the planar Cu sites and its significant discrepancy with the observed incommensurate neutron spin fluctuations. The theory is derived from the recently published ab initio band structure calculations that correct the tendency of LDA computations to overestimate the self-coulomb repulsion for the half-filled Cu d(x2-y2) orbitals for these ionic systems. The new band structure leads to two bands at the Fermi level with holes in the Cu d(z2) and apical O p(z) orbitals in addition to the standard Cu d(x2-y2) and planar O p(sigma) orbitals. This band structure is part of a new theory for the cuprates that explains a broad range of experiments and is based upon the formation of Cooper pairs comprised of a k up arrow electron from one band and a -k down arrow electron from another band (interband pairing model).