먹이다 is the causative of 먹다: to feed someone else rather than eat yourself. The -게 adverb says how — 한 끼 맛있게 먹이고 싶은 거지, 'we just want to feed you one good meal'. Two related compounds appear here as well: 먹여주다, feeding someone by hand, and 먹여살리다, supporting a household. The point to hold on to is that the eater is somebody other than the subject.