Klopp, who won the Premier League and Champions League at Liverpool, was known for developing his players both on and off the field.
Robertson told Premier League Productions after a final day win over Wolves at home: "We can't thank Jurgen Klopp enough and all the players could do today was try to win for him and we managed that to send him off on a high.
"Every time he stood up he would have got a massive cheer. He tried to take the limelight off him and it just wasn't possible. He will have to suck it up today. What he's done for the club is magnificent. We have a lot to be grateful for and thankful for. Today he can just focus on being with his family and the fans. We wish him and his family all the best for the future.
"On the pitch it's clear to see. He gave me a chance at Liverpool, signed me, made me the player I am and helped me win trophies. I came here as a relatively young man and I'm a lot more older and wised now but a lot of characteristics have been down to him. He texts me in good moments and in bad. He's improved all of us as people. That's what we're all grateful for. He always took time for our families and for us. That's what we'll miss the most. He's done amazing at this club. I think he needs to get off in the sun, have a beer in his hand and have a well deserved rest.
"We wanted to send him off like we have but of course there's a new era upon us. We're all excited and we have to be. We need to be on board. We will be ready to get on board with the new manager and help him as much as we can. This club demands success. We're excited for the new challenge. Let's have a couple of days rest, I'll go represent Scotland, then we'll be ready."