가족만 초대하다
만 after a noun means 'only' or 'just.' It limits the group or thing you are talking about. You can reuse this pattern with many nouns, such as 친구만, 물만, 오늘만. In the story, '가족만 초대하기로 했어요' means they planned to invite only their family.
저는 친구만 초대했어요.
I invited only my friends.
