Carroll signed for the Hammers until January after being told by Reds manager Brendan Rodgers that he could not guarantee the England international a regular starting place.
The Hammers target man told ESPN that he knew the time was right to leave Anfield.
He said: "I haven't got a fair chance, I don't think, at Liverpool, and I was really told that I should move on to a team where I'll play regular football.
"I knew Sam Allardyce wanted us here and told us I'd be playing games so for me it was a no-brainer coming here.
"I want to play football all the time. Every time there's a game I want to be involved and obviously I didn't get a chance there. A new manager came in and he had what he wanted to do and I wasn't part of his plans and for me to play football this was a chance for me to do that, so that's why I made the move."
The 23-year-old made his comeback from a hamstring injury with a 20-minute spell towards the end of the Hammers' 2-1 win at Queens Park Rangers last Monday.