To be finger licking good means that the food one just ate was very tasty. You literally lick your fingers so as not to waste even one spot of the tasty food! It is not just kids who do this, but grown-up adults too!
Note: generally the word licking is given as lickin' - with an apostrophe after 'n' and without the 'g'.
1. During the Thai food festival the hotel guests unanimously agreed that the traditional vegetables were finger lickin' good.
2. The restaurant was popular primarily because the desserts were finger lickin' good and not due to the main courses.
3. Sarita was considered by both family and friends to be a great cook as whatever she dished up was finger lickin' good.