Translation:My daughters sat down at the table and started eating.
That is too simplistic.
За + accusative - behind (direction) or for (as in "for the reason of", "because of"):
Он в тюрьме за воровство = He is in prison for theft.
За + instrumental - behind (location) or for (as in "for the purpose of"):
Я иду в магазин за молоком - I am heading to the shop to buy milk.
Also it is used if you are drinking TO something. ЗА здоровье! За вас! - TO health! TO you!
"За столом" is a location, "за стол" is a direction. With "сели" you use the direction, because there's movement from one position to another. You'd have to use "за столом" if you were talking about a static picture. "Они сидят за столом" "they are sitting at the table".