The big striker scored just six Premier League goals for the Reds after joining them from Newcastle in January 2011, and was sent out on loan to West Ham after Brendan Rodgers arrived in the summer.
And he believes that he was not given the opportunity to do his best at Anfield.
"I haven't really had a fair chance at Liverpool," Carroll told ESPN. "I was really told that I should move on to keep playing regular football, and obviously then Sam (Allardyce) wanted me here and told me I would be playing regular games.
"So, for me it was a no-brainer coming here and to play.
"Obviously, I didn't really get a chance there. A new manager came in and I wasn't part of his plans.
"For me to play football, that was my chance to do that. That is why I am here at Upton Park.
"I think I could fit into every system. I believe in what I can do. Obviously he [Rodgers] has his beliefs and he didn't think that.
"But for me, obviously, it was better to move on instead of waiting for my chance - which he told me might not come.
"Coming here, I have got a great chance to start games and score goals, which I haven't really had many of last season.
"I don't think I have got a point to prove, that is the wrong way to put it. I know what I can do and the ability I have got. I know what I am capable of.
"If other people don't believe it, that is fine. I just get on with it. But at the end of the day, I know what I can do."