I live alone since when I was 18 years old. Many of my friends live alone, too. For me, and for my friends, it's simply because our schools are in Seoul, the capital and the largest city in the nation. Over one-third of whole people in South Korea live in Seoul, or near Seoul. There's a old proverb in Korea saying "Send your horses to Jeju Island, and send your kids to Seoul". Like this, Koreans like living in the capital city. So they keep coming. They come and live alone. And they become isolated. That's what the modern families look like. There are some more reasons of these situations. It's not simply because of an old proverb.
Divorced couples and people who prefer living single are keep on increasing. Traditionally, the concept Korean society and culture were had been collectivism. But, as globalization, which is in fact Westernization, individualism is becoming the mainstream of the culture. Young people, who had not liked the collectivism, is becoming more personal and individual, and sometimes selfish. They don't want to be involved into some group without their own will.
Another reason is that it getting harder and harder to live in Seoul because the economy in Korea is crushing. 40 years ago, there was a tremendous economical improvement in Korea so called "the miracle of Han River". But, as it was too fast and people were not prepared, side effects of it are always following us. Nowadays, it is impossible to buy a house, have a car, and raise at least two kids without dept if you are just a normal salary man. It is so natural that the number of family members are decreasing, even to one, under these circumstances.
Final reason is related with the proverb mentioned above. There are too many people in Seoul. They are here because they study here, or work here. Because of that, there are so many small apartments near universities and workplaces one person can live in. Then, some people who are not even related with Seoul want to live here, because others do. Seriously, there are many people in Seoul without specific reason, just because they want to and others do.
I don't know why these things happens in other countries, but in Korea, that's what it is. There is a old song called "Hail to Seoul", and its final line of lyrics is this. "In this beautiful Seoul, I want to live in Seoul." But nowadays, Seoul is becoming a sad city having many kind of stories of lonely people.