This expression is from a KBS TV drama called '프로듀샤' and explained by a Korean native speaker Anna.
For other expressions from KBS 프로듀샤, please click http://www.laughingbeans.com/learn/class/299
This is usually said by moms to their children while having meals. I'm not sure if moms in other countries do that too, but in Korea, moms say this quite often to their young children.
꼭꼭: thoroughly. This adverb is rarely used alone. This is usually combined with 씹어먹다.
씹어먹어라: chew it. Original form is 씹어먹다(chew).
verb+어라 means you order others to do something.Let's see more examples of verb+어라. (Ordering someone to do something) Please note that this sounds a bit like a command and outright 반말, so usually someone who is in higher ranks than you or older than you would say this to you.
1) 신발을 벗어라. Take off your shoes. 신발을 벗(다)+어라
2) 물을 마셔라. Drink water. 물을 마시(다)+어라: 마시어라 -> 마셔라
3) 책을 읽어라. Read books. 책을 읽(다)+어라