A metamaterial microstructure consisting of a single-pole resonator and two rectangular open-ring resonators is designed. The broadband electromagnetically induced transparency (EIT) effect and slow-light effect of the flexible metamaterial structure are studied, and a transparent window with a maximum transmittance of 86.96%, a relative bandwidth of 64.14%, and a slow-light group delay of 3.20 ps are obtained. When the Fermi level of graphene increases from 0 eV to 1 eV, the light transmittance decreases from 86.96% to 28.9%, and the modulation depth is 66.7%. In addition, when the surface of the microstructure is covered with an analyte with a refractive index of 1.0-1.8 and a thickness of 5 μm, the refractive index sensing sensitivity of the microstructure is 77.93 GHz/RIU. This study may provide a new direction for the active regulation of broadband EIT-like effects and high-sensitivity refractive index sensing devices in the terahertz (THz) band.