Here the word "They" is a generalization. It is unlikely that all the people of Tyre and Sidon went to Herod. Alternate translation: "Men representing the people of Tyre and Sidon went with one purpose to talk with Herod" (See: [[rc://en/ta/man/translate/figs-hyperbole]])
They probably purchased this food. Alternate translation: "the people of Tyre and Sidon bought all their food from the people that Herod ruled" (See: [[rc://en/ta/man/translate/figs-explicit]])
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]])