The temperature and strain dependences of the optical conductivity spectrum of hexagonal manganese telluride (MnTe) were measured, revealing absorption in the terahertz (THz) region from spin-split bands to acceptor levels. The temperature dependence of the THz absorption peak is consistent with that of a ferromagnetic phase transition, even though MnTe exhibits no net magnetism. The temperature dependence was attributed to a change in the altermagnetic electronic structure. Under negative uniaxial strain, the THz peak shifts to the high-energy side, suggesting spin-splitting bands at energies away from EF, consistent with the theoretical prediction that the spin-splitting angle decreases. The observed behavior of the THz peak clearly shows that MnTe has the altermagnetic electronic structure. Additionally, a Fano-like asymmetric line shape in the optical phonon absorption was observed, possibly originating from interactions with the alternative spin-split bands.