The France international, who missed the first three games of the season as he was pursued by the London club, made his first start on Saturday as the Magpies beat Aston Villa 2-1 and received a warm ovation as he was substituted off.
"If the fans were - or still are - mad at me for what happened then I understand and I apologise to them. I want to come back from what has happened," Cabaye told the club's website.
"For Newcastle fans, football is their life and I want to say that I am going to give everything - as I have from my first day here - from today until the end of the season."
Last month, angry manager Alan Pardew said the former Lille player had been distracted by the £10m bid from Arsenal on the eve of Newcastle's Premier League opener at Manchester City, and branded the offer disrespectful and ill-timed.
Cabaye said he had moved on and was happy to stay at the club in an important year leading up to the World Cup.
"I want to forget what happened during the summer, I want to put it behind me and give everything for Newcastle every day, for the club and for my team mates," he said.
"It is very important to be right in my head and I am."
- Sports & Recreation