Two key ingredients to the transformation (maybe 3).
1. A coherent system instilled by Klopp (who - or his coaching team - also has the knowledge to make subtle adjustments to make it work). I think Rodgers has a system in mind, but never the ability to have in executed or to make necessary adjustments).
2. Self Confidence. It does not matter how good a player is, without confidence they becomes shadows of themselves (completely different example is when Messi plays for Argentina - nothing like the player at Barca). Klopp has had a huge impact on most here for Lovren, Moreno, Sahko, Can, Milner. Exceptions include Ibe, Skrtyl, and maybe Sturridge. I think Klopp is playing the long game with Sturridge. It may turn out brilliantly. We will either get a team player or worst case is that his current fitness and scoring rate will lead to a much larger transfer fee than we could have imagined 3 months ago.
3. Team Confidence. And, as with Messi example, it is not just self-confidence, but also confidence in the teammates and system. Klopp has completely changed the dynamic here and it is obvious in the "big" games. No fear.
But, Credit to the players that have responded and Lovren has to be near the top of that list.