We report the result of optical identifications of FIRST radio sources with the Hyper Suprime-Cam Subaru Strategic Program survey (HSC-SSP). The positional cross-match within 1 '' between the FIRST and HSC-SSP catalogs (i less than or similar to 26) produced more than 3600 optical counterparts in the 156 deg(2) of the HSC-SSP field. The matched counterparts account for more than 50% of the FIRST sources in the search field, which substantially exceed previously reported fractions of SDSS counterparts (i less than or similar to 22) of -30%. Among the matched sample, 9% are optically unresolved sources such as radio-loud quasars. The optically faint (i > 21) radio galaxies (RGs) show that the fitting linear function of the 1.4 GHz source counts has a slope that is flatter than that of the bright RGs, while optically faint radio quasars show a slope steeper than that of bright radio quasars. The optically faint RGs show a flat slope in the i-band number counts down to 24 mag, implying either less massive or distant radio-active galactic nuclei (AGNs) beyond 24 mag. The photometric redshift and the comparison of colors with the galaxy models show that most of the matched RGs are distributed at redshifts from 0 to 1.5. The optically faint sample includes the high radio-loudness sources that are not seen in the optically bright sample. Such sources are located at redshift z > 1. This study gives similar to 1500 radio AGNs lying at the optically faint end and high-redshift regime not probed by previous searches.