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. A Fano-like antisymmetric line shape in the optical phonon absorption was observed, which originates from the interaction between optical phonons and the spin-split bands. Additionally, under negative uniaxial pressure, the THz peak shifts away from the Fermi level (EF), suggesting that 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.