It's a figurative expression that compares the speed of something to the blink of an eye, which is a very fast action.
Here are some synonyms for "in the twinkling of an eye":
* In a flash
* In a jiffy
* In an instant
* Immediately
* Suddenly
Example:
* "The magician disappeared in the twinkling of an eye."
* "The news spread like wildfire, reaching everyone in the twinkling of an eye."