The former England international signed a deal with Bruce Arena's side in January after cancelling his contract with Roma, where he had not made an appearance all season.
Cole drew the ire of some football fans in the United States when, after joining Roma from Chelsea in 2014, he claimed he rejected the chance "relax on a beach" by moving to MLS.
The 35-year-old says the quote was taken out of context and is now determined to change the perception of some supporters by working hard for more success.
"I'm not going to come here and try to defend myself," he said at his official introduction as a Galaxy player, alongside fellow new signing Jelle van Damme.
"It was said. It was taken out of content because I was talking to the reporter before and he kind of said it to me: 'We're glad to have you at Roma, we didn't expect you to be here, we thought you'd go relax on the beach.'
"In the interview, I've said it, yes. It's come out my mouth. But of course you have to understand that I was at a new team and I have to kind of tell the fans that at Roma I was here to fight, I was there to influence and play in the Champions League in Europe.
"Of course, I hold my hands up. It was said. I have to deal with it. I hope I can kind of change the views of me coming here.
"Of course I'm a winner. I always want to win. I don't come here to sit on the beach, to rest. I'm here to play football and work hard. I'm not a diva, I'm not an egotistical guy that comes and thinks he's bigger than anyone. I'll come here and work as hard as I can."
Cole says he has spoken to former Chelsea team-mate Frank Lampard and Galaxy midfielder Steven Gerrard about the difficulties of adapting to America's top flight.
"I've spoken to them on many occasions [about] how difficult it is. I spoke to the manager how difficult it is with the travelling, the demands of the game over here. I know its going to be hard for me for sure," he added.
"But I'm happy to be here. I'm going to work as hard as I can, change a few views of me being here, and we'll see."