Neutral and charged excitons (trions) in atomically-thin materials offer important capabilities for photonics, from ultrafast photodetectors to highly-efficient light-emitting diodes and lasers. Recent studies of van der Waals (vdW) heterostructures comprised of dissimilar monolayer materials have uncovered a wealth of optical phenomena that are predominantly governed by interlayer interactions. Here, we examine the optical properties in NbSe$_2$ - MoSe$_2$ vdW heterostructures, which provide an important model system to study metal-semiconductor interfaces, a common element in optoelectronics. Through low-temperature photoluminescence (PL) microscopy we discover a sharp emission feature, L1, that is localized at the NbSe$_2$-capped regions of MoSe$_2$. L1 is observed at energies below the commonly-studied MoSe$_2$ excitons and trions, and exhibits temperature- and power-dependent PL consistent with exciton localization in a confining potential. Remarkably, L1 is very robust not just in different samples, but also under a variety of fabrication processes. Using first-principles calculations we reveal that the confinement potential required for exciton localization naturally arises from the in-plane band bending due to the changes in the electron affinity between pristine MoSe$_2$ and NbSe$_2$ - MoSe$_2$ heterostructure. We discuss the implications of our studies for atomically-thin optoelectronics devices with atomically-sharp interfaces and tunable electronic structures.