The magnetic proximity effect is a fundamental feature of heterostructures composed of layers of topological insulators and magnetic materials since it underlies many potential applications in devices with novel quantum functionality. Within density functional theory we study magnetic proximity effect at the three-dimensional topological insulator/magnetic insulator (TI/MI) interface in the Bi₂Se₃/MnSe(111) system as an example. We demonstrate that a gapped ordinary bound state caused by the interface potential arises in the immediate region of the interface. The gapped topological Dirac state also arises in the system owing to relocation to deeper atomic layers of topological insulator. The gap in the Dirac cone originates from an overlapping of the topological and ordinary interfacial states. This result being also corroborated by the analytic model, is a key aspect of the magnetic proximity effect mechanism in the TI/MI structures.