According to the ancient meaning, your family left and someone came in.
Now, of course, it is superstition.
More scientific evidence is that you have a party at home recently, or mentioned the party problem, and your personal feelings have entered a low ebb (the implied thing may not be a cat, but also a plastic bag).
The result is that you long for your family, so you dream about your whole family and your dead relatives in your dreams.
As for the white couplets written by your grandfather, it is to remind yourself that this is a dream, and people will have white couplets when they die (there will be some details in any dream that make you think it is not true)
As for red couplets, it is an extension of white couplets, probably because your family used to get together and dream about the whole family during the New Year, and these festivals naturally have red couplets.
It doesn't mean much, just an extension of your subconscious feelings.