살리다 is the causative of 살다 — keep alive. Every line here is the fixed compound 먹여 살리다, literally 'feed and keep alive', meaning to support a household or a payroll: 열심히 일해야 직원들도 먹여 살리고 'you have to work hard to keep your staff fed', 요즘엔 애가 우리를 먹여 살리는 거야 'these days the kid is the one supporting us'. It always takes a person as its object, and it is often said half-jokingly about who the real breadwinner is.