Superconductivity in van der Waals materials, such as NbSe2 and TaS2, is fundamentally novel due to the effects of dimensionality, crystal symmetries, and strong spin-orbit coupling. In this work, we perform tunnel spectroscopy on NbSe2 by utilizing MoS2 or hexagonal boron nitride (hBN) as a tunnel barrier. We observe subgap excitations and probe their origin by studying various heterostructure designs. We show that the edge of NbSe2 hosts many defect states, which strongly couple to the superconductor and form Andreev bound states. Furthermore, by isolating the NbSe2 edge we show that the subgap states are ubiquitous in MoS2 tunnel barriers but absent in hBN tunnel barriers, suggesting defects in MoS2 as their origin. Their magnetic nature reveals a singlet- or a doublet-type ground state, and based on nearly vanishing g factors or avoided crossings of subgap excitations, we highlight the role of strong spin-orbit coupling.