。
A family is moving in next week.
Literal
Next-week [become-になる-when-と] one-family [subject-が] enter-come.
~になると ('when it becomes X') is a temporal conditional frequently used for time arrivals — '[when we] get to next week.' 入ってくる = 入る ('enter') + くる ('come'), describing motion toward the speaker's perspective. Often implies moving into a building or household.