A sword can only kill one person at a time and tends not to be very persuasive in the long run.
A pen can tell many people with many swords to kill many people, or turn many citizens against their rulers.

Obviously none of you hae been stabbed with a pen! Small, precise, fits between your ribs. Let's just say I would have rather had a clean sword slice than the jagged pen stabbing I got.

Hmmm...I'm not sure your analogy serves, as neither a pen nor a sword is an action or word. Both are objects - nouns.

The act of writing is an action, it is expressing an idea. A sword can harm bodies, but the written word can change hearts and minds.

You use this proverb to say that you can solve problems or achieve your purpose better and more effectively through communication with words than by violence with weapons.

The written word is a powerful thing. Sometimes, spoken words disappear in the air as if they were vapor. The written word can stand out as a bold witness years later. Ready to speak again when the reader comes across it.
By spoken word, civilizations have passed down their legends and thus stayed alive for centuries.
The civilizations that we know best though are the ones that have had their story written down and preserved for us.

Very deep question!
If we were gonna turn this into a sort of rock, paper, scissors, I'd take the "A picture is worth a thousand words!" to block.

Many here have made GOOD points, Action IS important, but a formal declaration of war can kill many more than a single sword.

But proof of an atrocity(in a picture) can move many to action faster than someone telling about it in writing.

Actions when timed correctly, can turn the tide of revolution or war. Take Hitler and Napoleon and their invasions into Russia. Had they timed it better both may have had success with dire consequences. The pen on the other hand symbolizes an idea or concept whose time has come. Thats why poets, writers, and other intellectuals are murdered during revolutions and times of civil unrest. Its also why books and documents are burned or censored. Ideas created by the pen can be very detrimental to a dictator or anyone who struggles to control great masses of people through fear and intimidation.

The phrase "the pen is mightier than the sword" can have many meanings. I like to take the example of the Declaration of Independance. Americans fought long and hard for their freedom with guns and swords. But the mightiest act of all came when a group of men got together and wrote the Declaration of Independance. In this document, America's forefathers were making a strong statement against England. With that declaration, they declared their intent to be a freed from the crown. That document did much to attain freedom for New England. Because of that document, America is a free country. The hearts and souls of those under the rule of the British crown were poured out with that pen. And thus America was born, not because of the might of the sword (or gun), but because what was poured out onto that parchment with that pen was the might, strength, and desires of a people fighting to be free.

In addition to Everest's excellent answer, if you think about what you have achieved in your life. Education, qualifications, a job, marriage, the ability to drive, penpals, even those people you send CHirstamd cards to but never manage to meet up with. How much of that could you accomplish with a sword rather than a pen?

Words can change the way people think and even change their hearts. Swords and other weapons can only restrain, intimidate, and control while they are being brandished. Words can effect permanent change; swords are temporary "fixes" at best.