The UDB rearranges verses 4 and 5 into a verse bridge in order to make it easier to shorten the sentences. (See: [[rc://en/ta/man/jit/translate-versebridge]])
The phrase "the city of David" was a name for Bethlehem that tells why Bethlehem was important. Although it was a small town, King David was born there, and there was a prophecy that the Messiah would be born there. Alternate translation: "to Bethlehem, the city of King David" or "to Bethlehem, the town where King David was born" (See: [[rc://en/ta/man/jit/figs-explicit]])