We use Z-contrast imaging and atomically resolved electron energy-loss spectroscopy on an aberration-corrected scanning transmission electron microscope to investigate the local electronic states of boron atoms at different edge structures in monolayer and bilayer hexagonal boron nitride (h-BN). We find that edges with bonding unsaturated sp² boron atoms have a unique spectroscopic signature with a prominent pre-peak at ~ 190.2 eV in the B K-edge fine structure. First-principles calculations reveal that the observed pre-peak arises from excitations to the in-plane lowest-energy empty sp² boron dangling bonds at the B-terminated edge. This spectroscopic signature can serve as a fingerprint to explore new edge structures in h-BN.