What is the meaning and reasoning behind the Open/Closed Principle?

It means that you should put new code in new classes/modules. Existing code should be modified only for bug fixing. New classes can reuse existing code via inheritance.

Open/closed principle is intended to mitigate risk when introducing new functionality. Since you don't modify existing code you can be assured that it wouldn't be broken. It reduces maintenance cost and increases product stability.

Specifically, it is about a "Holy Grail" of design in OOP of making an entity extensible enough (through its individual design or through its participation in the architecture) to support future unforseen changes without rewriting its code (and sometimes even without re-compiling **).

Some ways to do this include Polymorphism/Inheritance, Composition, Inversion of Control (a.k.a. DIP), Aspect-Oriented Programming, Patterns such as Strategy, Visitor, Template Method, and many other principles, patterns, and techniques of OOAD.

** See the 6 "package principles", REP, CCP, CRP, ADP, SDP, SAP