"to separate the day from the night." This means "to help us tell the difference between day and night." The sun means it is daytime, and the moon and stars mean it is nighttime.
This is a command. By commanding that they should serve as signs, God made them serve as signs. AT: "Let them serve as signs" or "let them show" (See: [[rc://en/ta/man/translate/figs-imperative]])
"It happened like that" or "That is what happened." What God commanded happened just as he said it should. This phrase appears throughout the chapter and has the same meaning. See how you translated this in [Genesis 1:7](./06.md).