살리다 is the causative of 살다: keep something alive, rescue it, or bring it back — a business (업무 협약 맺고 살려낸 건데), an economy (살려야 우리나라 경제에도 좋고), a person you are begging (바짓가랑이를 잡고 살려만 달라!). The -고 in front simply chains the preceding action: 잡고 살려주세요 "grab on and beg to be saved." Note the cheer 살리고 살리고, shouted to mean "keep it up, keep it alive," and 살려주세요, the standard cry for help or mercy.