It's a somewhat outdated phrase, but it still gets used occasionally, especially in informal settings.
Here's why it's called that:
* In the past, when someone was fired, they were often given a piece of paper with their final paycheck and a written explanation of their dismissal. This paper was sometimes called a "walking paper" because it literally allowed the employee to "walk out" of the company.
Here's an example of how it might be used:
"The boss was really angry about the missed deadline, so he gave her walking papers."