The idea
Carl Gustav Jung, the Swiss psychiatrist, used the word synchronicity for meaningful coincidence: two events that don't cause each other, yet for the person living them are linked by meaning. He used the term publicly in 1930, in his memorial address for Richard Wilhelm, the German translator of the I Ching, and developed it in a 1952 essay written in dialogue with the physicist Wolfgang Pauli.
The I Ching and Jung
Jung consulted the book for decades. In his foreword to the 1950 English translation, he suggested that the fall of the coins and the state of the person asking coincide in the same moment, and that the text that comes up can be meaningful through that coincidence, not through a cause. That foreword led many Western readers to take the book seriously, and the idea is still one of the most common ways people explain why a reading seems to "hit."
What isn't proven
Synchronicity isn't a scientific finding. No experiment has shown that coins reflect the state of the person asking. Psychology offers other explanations for that feeling of a hit: we tend to see patterns, open-ended texts fit many situations, and simply thinking calmly about a question already clears up a lot.
How to use it
You don't need to believe in synchronicity for the I Ching to be useful. You can treat it as a mirror: an old text, chosen by chance, that makes you look at your situation from another angle. More in Chance and Meaning.