Here's why:
* Deer are known for their agility and speed. They can run at speeds up to 40 mph, making them difficult to catch.
* The idiom is often used in situations where someone needs to escape danger. For example, "He saw the police coming and ran like the deer."
Other ways to say it:
* Run like the wind
* Bolt
* Dash
* Make a run for it
* Take off like a shot
The idiom suggests a sense of urgency and speed, similar to how a deer would flee from a predator.