Scanning tunneling microscopy (STM) by providing images of the effects of individual zinc impurities in cuprate superconductors with unprecedented atomic resolution offers a stringent test to models of correlated fermions for high-temperature superconductors. Using a t-J model supplemented by variational Monte Carlo many-body techniques, the spatial dependence of the hole density and of the valence bond and superconducting pairing amplitudes around the impurity are computed. Our calculation supplemented by the theory of interlayer tunneling by I. Martin [Phys. Rev. Lett. 88, 097003 (2002)] can explain most features of the STM experiments including the cross-shaped spatial structure and the impurity "resonant" peak.