Magnon-polaritons are shown to play a dominant role in the propagation of terahertz (THz) waves through TmFeO₃ orthoferrite, if the frequencies of the waves are in the vicinity of the quasi-antiferromagnetic spin resonance mode. Both time-domain THz transmission and emission spectroscopies reveal clear beatings between two modes with frequencies slightly above and slightly below this resonance, respectively. Rigorous modeling of the interaction between the spins of TmFeO₃ and the THz light shows that the frequencies correspond to the upper and lower magnon-polariton branches. Our findings reveal the previously ignored importance of propagation effects and polaritons in such heavily debated areas as THz magnonics and THz spectroscopy of electromagnons. It also shows that future progress in these areas calls for an interdisciplinary approach at the interface between magnetism and photonics.