The word catholic comes from the Greek word “(katholikos from katholou — throughout the whole, i.e., universal)“, you can read the entire article here: NewAdvent.org
You must remember that at that time shortly after the Apostles died, there were “new churches” popping up everywhere and many taught heretical doctrines. What is old is new again I suppose.
But the best short version of how Christians came to be called Catholic is a quote from St. Pacian from the 4th Century.
“Christian is my name, and Catholic my surname. The one designates me, while the other makes me specific. Thus am I attested and set apart…When we are called Catholics it is by this appellation that our people are kept apart from any heretical name.”