Vapor condensation on weakly interacting substrates leads to the formation of three-dimensional (3D) nanoscale islands (i.e., nanostructures). While it is widely accepted that this process is driven by minimization of the total film/substrate surface and interface energy, current film-growth theory cannot fully explain the atomic-scale mechanisms and pathways by which 3D island formation and morphological evolution occurs. Here, we use kinetic Monte Carlo simulations to describe the dynamic evolution of single-island shapes during deposition of Ag on weakly interacting substrates. The results show that 3D island shapes evolve in a self-similar manner, exhibiting a constant height-to-radius aspect ratio, which is a function of the growth temperature. Furthermore, our results reveal the following chain of atomic-scale events that lead to compact 3D island shapes: 3D nuclei are first formed due to facile adatom ascent at single-layer island steps, followed by the development of sidewall facets bounding the islands, which in turn facilitates upward diffusion from the base to the top of the islands. The limiting atomic process which determines the island height, for a given number of deposited atoms, is the temperature-dependent rate at which adatoms cross from sidewall facets to the island top. The overall findings of this study provide insights into the directed growth of metal nanostructures with controlled shapes on weakly interacting substrates, including two-dimensional crystals, for use in catalytic and nanoelectronic applications.