'I got the word out means' I spread the word, I let people know what was happening, I made a particular piece of news public, I informed as many people as possible etc

I got the word out = I told everybody (The information is out, and is there for anybody to see) The word (information) is out.

It means that you have told people information about something

To make something known.

the word is out is like finally we know the answer to something we want to know i got the word out is you are the one who is telling this information