Actually the basic meaning of "챙기다" is "do not make it missed" or "keep in mind" So, 밥 잘 챙겨먹어 has the meaning of "Don't miss your meal." or "Keep your meal in mind" 동생들 잘 챙겨 has the meaning of "Keep your brothers in mind"

챙기다 has various meanings 밥을 챙기다 means take 동생을 챙기다 means take care of