Meaning of: "No man is an island, entire of itself"

No, it's not a proverb. It's a quotation, and it was written not by Francis Bacon, but by John Donne. It’s from his Meditation XVII. Its meaning is perhaps apparent in the continuation, closing with the famous words that gave Ernest Hemingway the title of one of his books:

No man is an island, entire of itself; every man is a piece of the continent, a part of the main. If a clod be washed away by the sea, Europe is the less, as well as if promontory were, as well as if a manor of thy friend's or of thine own were. Any man's death diminishes me, because I am involved in mankind; and therefore never send to know for whom the bell tolls; it tolls for thee.

It means human can not get along with their lives alone and succeed, and we are all dependent on other people and we need them. We can't thrive just by being alone.

It means that since we live in a cooperative and interdependent society, your actions will always affect someone else. The only way for your actions to affect no one but yourself is to live alone on an island.