There has been a long-standing debate about the mechanism of the unusual superconductivity in alkali-intercalated fullerides. In this Letter, using high-resolution angle-resolved photoemission spectroscopy, we systematically investigate the electronic structures of superconducting K_{3}C_{60} thin films. We observe a dispersive energy band crossing the Fermi level with the occupied bandwidth of about 130 meV. The measured band structure shows prominent quasiparticle kinks and a replica band involving the Jahn-Teller active phonon modes, which reflects strong electron-phonon coupling in the system. The electron-phonon coupling constant is estimated to be about 1.2, which dominates the quasiparticle mass renormalization. Moreover, we observe an isotropic nodeless superconducting gap beyond the mean-field estimation (2Δ/k_{B}T_{c}≈5). Both the large electron-phonon coupling constant and large reduced superconducting gap suggest a strong-coupling superconductivity in K_{3}C_{60}, while the electronic correlation effect is suggested by the observation of a waterfall-like band dispersion and the small bandwidth compared with the effective Coulomb interaction. Our results not only directly visualize the crucial band structure but also provide important insights into the mechanism of the unusual superconductivity of fulleride compounds.