This quotation phrase, No man is an Island ,from the understanding of islands that being self sufficient and independent.It is compared with men because no matter how able of them, no person can do everything by themselves.
Human being are social animals and cannot do by ourselves independently no matter how we want to be independent and live and stand up on our own unlike the islands do.
Literally means, though island cannot do by itself because it cannot move by itself, human beings are not the same as an island, we people tend to have companions to help for us, to care for us because that is all we need. 
We always need help every now and then because we cannot live at all times, even a robots need help to do their tasks with humans.
We says that we are independent because we are now have a stable job but basically there is always times that we usually need our family for their support, we may be could live physically independent but we need them
emotionally, that is how it works, so there is no man is an island at all.
So from my understanding, we individuals do not just exist entirely being alone and do by ourselves, but we all connected to everyone, therefore we are part of the island and we have to do thw work that the island cannot, it is a mutual understanding.
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