Actions do speak louder because people say a lot of words, sometimes even lies, but actions that they do in their life show their true face.
Pen is mightier than the sword because sword leads only to one thing, and that is death and destruction.
Pen on the other hand gave us world knowledge. It gives us insight into nature and society. Into imagination and fantasy.
There is no doubt that sword's do have great influence on history trough birth and death of empires, kingdoms, countries. But no one would actually know about these "accomplishment" of the sword without pen to write it down.
Such a thing is certainly an action.
Consider the action of Karl Marx writing the Communist Manifesto, or the action of Darwin writing the Origin of Species. These actions certainly spoke louder than mere words.
These actions changed the world as we knew it.
That is why the pen is mightier than the sword.