“this day.” The Jewish day began at sunset; hence the angel was telling them what had happened sometime after sunset that evening. Jesus had been born after sunset.
“in the city of David.” The angels could have said, “in Bethlehem,” and been perfectly accurate, so why call Bethlehem, “the city of David” in this instance? The angels were announcing the birth of the long awaited Messiah, and the mention of David’s name and his ancestral home also brought back to mind all the wonderful Messianic prophecies spoken about the Messiah by David and the prophets. The Psalms of David are full of Messianic prophecies and references to the Messiah, and no doubt many of them would have been the subject of discussion as the shepherds walked (hurriedly walked) from the fields into the town of Bethlehem.
“the Savior.” We have translated this with “the,” although the Greek lacks the definite article. As Lenski says, “The relative clause [“who”] makes ‘Savior’ definite.”
“Messiah and Lord.” These words function like adjectives in the Greek, describing the Savior (Cp. Lenski). These adjectives are descriptive of the baby, showing that he has both the properties of being the Messiah and the Lord. To translate the phrase as, “who is Messiah the Lord,” misses this point. We use the term Messiah here instead of “Christ” to make it clear that what the angels were saying. The angels were expressing that the “Messiah,” the “Anointed One” had been born, and the text needs to make that clear.