Zouma pleaded guilty to kicking and slapping his pet and was sentenced to 180 hours of unpaid community work. He has also been banned from owning cats for five years.
"It's been a difficult spell for me and my family," he told Sky Sports. "Obviously I have done something very bad and I apologise for what I have done.
"I know it was very tough for people to watch and see that and obviously I feel very, very sorry and I have great remorse about it. Now I am trying to move on with my family and I'm looking ahead.
"I learnt from it, that is the most important thing. Obviously I have support from many people who helped me stay focused on football and obviously the club has been unbelievable with me.
"The support that I receive from my team-mates from the staff, everyone at the club and even from the fans, it's been unreal.
"They helped me stay focused on the game, trying to do my best on the pitch all the time. Obviously I can't thank them enough for what they have done. This club is like my family."
Zouma was fined by West Ham following the incident, with the money having been donated to nine different animal welfare charities.
The French defender has featured in 13 games for the Hammers this season, scoring one goal.