We report on the observation of high-temperature (Tc) superconductivity and magnetic vortices in single-unit-cell FeSe films on anatase TiO2(001) substrate by using scanning tunneling microscopy. A systematic study and engineering of interfacial properties has clarified the essential roles of substrate in realizing the high-Tc superconductivity, probably via interface-induced electron-phonon coupling enhancement and charge transfer. By visualizing and tuning the oxygen vacancies at the interface, we find their very limited effect on the superconductivity, which excludes interfacial oxygen vacancies as the primary source for charge transfer between the substrate and FeSe films. Our findings have placed severe constraints on any microscopic model for the high-Tc superconductivity in FeSe-related heterostructures.