The midfielder has signed an 18-month contract after leaving Liverpool and is available to face Manchester United in the FA Cup on Saturday.
Cole started his career at West Ham but left them to join Chelsea in 2003 and he says he is delighted to be back.
"It is a great move for me," the 31-year-old said. "Coming back here brings back all the memories of the pitches I learnt how to play on. This is where I learnt my trade and I'm happy to have another chance to be here. I've always loved West Ham and I hope I can give the club a few memorable seasons.
"The timing was right for me to come back and I'm happy. I'm back in London and I'm delighted to be here. I want to be here and I want to make an impact.
"The place has not changed much to look at, but I think Sam has done a great job with the team. A few years ago, the club was in a bad way and Sam has done brilliantly. All in all, it will always be West Ham and the history will always be here and everywhere I go around the world, people love West Ham. Everywhere I go, people know me for having started here."
The midfielder has left Anfield after a troubled two-and-a-half-year spell following his transfer from Chelsea in July 2010. He made only 42 appearances for Liverpool, scoring five goals – the last of which was against West Ham in December. He said the chance to play regular first-team football under Sam Allardyce was a major factor in his return to the Boleyn Ground.
"Sam didn't have to sell me the club for me to come back," Cole said of the West Ham manager. "For me, I have had 18 months at Liverpool where I haven't played much football and Sam wants to play me. I'm going to need games to get back to my best, but I want to play football and come here and help the team to make an impact.
"I could see from the outside what a great bunch of lads we have here and there is a great spirit. I'm excited."
Cole has said he is keen to impress on his return to his boyhood side. "I feel like I owe the club a little bit as well because if it wasn't for West Ham I wouldn't be a footballer. This club's great youth-team coaches taught me the trade."
Cole spent much of last season on loan at the French club Lille, scoring nine goals in 42 appearances. "We were a Champions League team and finished third in the league and it was a good technical league with a lot of great players. I played 40-odd games and enjoyed it. It was only during my time at Liverpool that I didn't play as much as I'd have liked. That happens, but now I've moved on and hopefully I'll have a great time here."
The midfielder was 19 when West Ham pulled off a memorable 1-0 FA Cup win over Manchester United in January 2001 and he believes his new club can beat them again when they face the Premier League leaders on Saturday. "We can do it again. It's a different team and a different era but at the end of the day I've come here to play football, I love playing football, and what better way to do it than against Manchester United at home.
"I'm just going to go out there, like I've always done, and give it my all and we have enough quality experienced players here to do something special."
West Ham have made an approach to take the Arsenal striker Marouane Chamakh to Upton Park, manager Sam Allardyce said. "We have made an inquiry and that is all we have done at the moment," said Allardyce.
"From my point of view, the situation I have with Andy Carroll being injured and Modibo Maiga going to the African Nations Cup, means as a striker we have Carlton Cole and that is it until Andy gets fit.
"If it is a possibility that it could be a deal done, then we are looking at it, as well as a number of others."