Klostermann, who was born in Germany, started his professional career with the youth academy of Eintracht Frankfurt before making his debut for their first team in 2014. He made his Bundesliga debut on August 31st, 2016, against VfB Stuttgart.
Klostermann quickly established himself as a key player for Eintracht Frankfurt, helping them win the German Cup in 2017. In 2018, he moved to Borussia Dortmund, where he played under manager Thomas Tuchel. Under Tuchel, Klostermann developed into one of the most important players for Dortmund,Bundesliga Tracking helping them win the Bundesliga title in 2019-2020.
In January 2021, Klostermann signed for RB Leipzig, becoming the club's new captain. Since joining Leipzig, Klostermann has continued to develop as a player, helping the team win the Bundesliga title in 2021-2022. He has also been instrumental in the team's success in European competitions, including reaching the UEFA Champions League semi-finals in 2022.
Overall, Klostermann's time at RB Leipzig has been marked by consistent performance and progress. He has become a key player for the team, contributing to their success both domestically and internationally. With his talent and dedication, it is clear that Klostermann will continue to be a key player for RB Leipzig for years to come.