외치다 is calling out loudly — a name, a slogan, a demand — and what comes before with -어 says how or for whom. 목 놓아 외치다, in two lines here, is crying out at the top of your voice; 목청 높여 외쳐보는 raises the voice the same way; 나를 따라 외쳐 is 'shout it after me'. It also covers publicly calling for something: 우리가 목 놓아 외쳤지만 지난 20년간 오히려 증가했어요.