Here the word "They" is a generalization. It is unlikely that all the people of Tyre and Sidon went to Herod. AT: "Men representing the people of Tyre and Sidon went together to talk with Herod" (See: [[rc://en/ta/man/translate/figs-hyperbole]])
# their country received its food from the king's country
They probably purchased this food. AT: "the people of Tyre and Sidon bought all their food from the people that Herod ruled" (See: [[rc://en/ta/man/translate/figs-explicit]])
# received its food
It is implied that Herod restricted this supply of food because he was angry with the people of Tyre and Sidon. (See: [[rc://en/ta/man/translate/figs-explicit]])