The 28-year-old agreed personal terms on Monday after being released by Chelsea at the end of last season and then flew to the club's training camp in Switzerland to undergo his physical examination before penning a four-year deal.
Much was made of the wage demands - reportedly £90,000-a-week - the England international was making having become a free agent following his release from Chelsea.
Cole said he remembered the atmosphere when he played at the ground for Chelsea in the Champions League semi-final of 2005, which Liverpool won on penalties before going on to lift the trophy in Istanbul, and was looking forward to experiencing that on a more regular basis.
"This is a massive club. I tried to take everything out of the equation, take the financial and location side out and just thought in football terms," said the England international.
"I thought about the semi-final of the Champions League in 2005 when I ran onto the field and the hairs on the back of my neck stood up.
"I was thinking about playing in that atmosphere every week and that swung it for me.
"I know I have made the right decision and I am looking forward to the challenge.
"I have played in London all my life. I could have stayed at Chelsea because the fans loved me and I won things, but I wanted to challenge myself and when I knew Liverpool were interested it was a no-brainer because they are the biggest club in the country."