어서 없애다

get rid of it by X

An "-아서/-어서" clause naming the method, plus 없애다 "get rid of, eliminate". Each line pairs a way with a removal: eat the leftovers up fast and they're gone, set traps and get rid of the cockroaches, incinerate it to dispose of it, scrape the flesh off to remove it, so many bedbugs came out that they had them exterminated. It also covers abolishing things — removing foreign students on national-security grounds. 없애다 is causative: you make something cease to exist, where 없어지다 is it going away on its own.