Stewart has made five appearances for the Reds since returning from a loan spell at Swindon Town.
Stewart was one of several players who recalled from his loan spell to train with side at Melwood, and was given a few opportunities in the FA Cup as Klopp made several changes in order to guide the club through a congested fixture list during an injury crisis.
The midfielder has told the official Liverpool website: "I haven't known him for a long time but I think he made an instant impact on me the first time I met him.
"The game I played at Exeter, his pre-match talks and the way he dealt with players, I found it perfect for me. He tries to take the pressure off of you as younger players playing for the first team.
"He just reminded us that it's just a game, not putting too much pressure on ourselves. [He encouraged us to] go out there and express ourselves and if anything goes wrong he's happy to take responsibility for it.
"He got us in the mentality to go out there and just play as we normally do for an U21s game."
Stewart certainly seemed to thrive in that environment, impressing in all four of his appearances in the FA Cup, and was given an opportunity in the Premier League as he featured as a second half substitute in the 6-0 win over Aston Villa last month.
Unfortunately an ankle injury means he has had to miss the past few weeks but Stewart was awarded a new deal for his efforts and looks as if he could be a squad player going forward.
Stewart has said he is progressing well and could be available for selection again very soon, adding: "I didn't expect it to get better so quick but I've been out on the pitch for about a week now doing some running and passing.
"So it's gone a lot quicker than expected."