Why can’t they hear me? I’d better sing even more loudly.
Literally: release – song – high – recite / chant / sing
Alternately: Singing in a loud voice, without worrying about whether you may be bothering anybody else.
This is a compound of compounds, and its parts may be flipped to produce 高吟放歌; unlike many such compounds, the first halves of the parts may also be swapped, to produce 高歌放吟 (kouka hougin). Both of these variants seem to be significantly less common than the original order (presented above), though.
This phrase is surprisingly modern: it is attributed to a 1948 short story titled 「テニヤンの末日」 (Teniyan no matsujitsu), by 中山義秀 (Nakayama Gishuu).