Carroll was farmed out on loan to West Ham after Brendan Rodgers replaced Kenny Dalglish.
But the 23-year-old feels he could have adapted his style to suit Rodgers.
"I haven't really had a fair chance at Liverpool," Carroll said. "I was really told that I should move on to keep playing regular football, and obviously then Sam 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."