Reconfigurable intelligent surfaces (RISs) not only assist communication but also help the localization of user equipment (UE). This study focuses on the indoor localization of UE with a single access point (AP) aided by multiple RISs. First, we propose a two-stage channel estimation scheme where the phase shifts of RIS elements are tuned to obtain multiple channel soundings. In the first stage, the newtonized orthogonal matching pursuit algorithm extracts the parameters of multiple paths from the received signals. Then, the LOS path and RIS-reflected paths are identified. In the second stage, the estimated path gains of RIS-reflected paths with different phase shifts are utilized to determine the angle of arrival (AOA) at the RIS by obtaining the angular pseudo spectrum. Consequently, by taking the AP and RISs as reference points, the linear least squares estimator can locate UE with the estimated AOAs. Simulation results show that the proposed algorithm can realize centimeter-level localization accuracy in the discussed scenarios. Moreover, the higher accuracy of pseudo spectrum, a larger number of channel soundings, and a larger number of reference points can realize higher localization accuracy of UE.