Robertson, 24, joined Liverpool from Hull City 18 months ago in a £10million deal.
But his early days were tough with Klopp sidelining him and giving him limited opportunities to shine.
Robertson started just two Premier League games for Liverpool before December last year leading to the Scot to grow annoyed.
“I wouldn’t say that I doubted myself but maybe I got a bit frustrated because you are used to playing every week and then when that stops it’s hard,” Robertson said.
“I could completely understand why I didn’t travel to games but it doesn’t make it any easier.”
But the turning point was Robertson’s performance in a 5-1 win at Brighton in December.
The left-back did enough to impress his boss and has been virtually an everpresent since, turning himself into one of the fans’ favourites.
“I finally got the chance to prove to everyone that I could make the step and I think I’ve done it,” he said.
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“The last year I look back upon with fond memories. I’ve loved every minute of it, loved coming into training every day, loved playing in all of the games I’ve been involved in.
“Coming from a relegated club, I don’t think there was as much excitement about me signing as there was with Mo Salah in the same window!
“But I wanted to prove that I could make the jump.”
Robertson is once again set to start for the Reds against Bournemouth today as they continue to their title challenge.
The Reds are two points behind Manchester City at the top of the Premier League having not lost all season.
Robertson added: “We can’t take our eye off one competition to concentrate on the other. The title race looks to have opened up a bit between us and Man City and the rest but we know that Tottenham, Chelsea, Man United and Arsenal will still all be there.
“The midweek game is a big challenge because we know that if we want to progress then we have to win the game. We know the different scenarios but we will look to just win the game and the rest deals with itself.
“We’ve played Napoli before, we know how good they are. But in the Champions League we’re lucky to have a special atmosphere. The Man City game at home (last season) was special, the Roma game was special, and the first game of this campaign against PSG was special.
“We will rely on our fans to give us the boost we need and I’m sure they will because they’ll create a special atmosphere and I’m sure the boys will react to it like they usually do.”