Heterostructures based on atomically thin semiconductors are considered a promising emerging technology for the realization of ultrathin and ultralight photovoltaic solar cells on flexible substrates. Much progress has been made in recent years on a technological level, but a clear picture of the physical processes that govern the photovoltaic response remains elusive. Here, we present a device model that is able to fully reproduce the current-voltage characteristics of type-II van der Waals heterojunctions under optical illumination, including some peculiar behaviors such as exceedingly high ideality factors or bias-dependent photocurrents. While we find the spatial charge transfer across the junction to be very efficient, we also find a considerable accumulation of photogenerated carriers in the active device region due to poor electrical transport properties, giving rise to significant carrier recombination losses. Our results are important to optimize future device architectures and increase power conversion efficiencies of atomically thin solar cells.