We report an in situ scanning tunneling spectroscopy study of one-unit-cell (1-UC) FeSe film on SrTiO 3 (001) (STO) substrate. In quasiparticle density of states, bosonic excitation mode characterized by the "dip-hump" structure is detected outside the larger superconducting gap with energy comparable with phonon and spin resonance modes in heavily electron-doped iron selenides. Statistically, the excitation mode, which is intimately correlated with superconductivity, shows an anticorrelation with pairing strength and yields an energy scale upper-bounded by twice the superconducting gap coinciding with the characteristics of magnetic resonance in cuprates and iron-based superconductors. The local response of tunneling spectra to magnetically different Se defects all exhibits the induced in-gap quasiparticle bound states, indicating an unconventional sign-reversing pairing. These results support the magnetic nature of the excitation mode and possibly reveal a signature of electron-magnetic-excitation coupling in high-temperature superconductivity of 1-UC FeSe/STO.