I am for the first time in Stuttgart and I saw these things and I am not sure what they are ? What are their purpose? Can someone help me out to figure out that?
These are, basically, umbrellas/parasols. If you Google for "Liederhalle Schirme" ("Schirm" being kind of a generic word for sun/rain protection) you will find the website https://www.arneggergmbh.de/de/schirme.html, which apparently is the company who produced these things - they have little images of their products on their website, and the respective image file is named "liederhalle.jpg" (top row, right), which suggests that these are the very same items.
More specifically, these are "Trichterschirme" ("Trichter" being "funnel"), meaning that the rain water is caught in the umbrella and drains through the hollow base.
These things are located at the entrance of the Liederhalle, "an internationally renowned culture and convention centre". It's functional art.
Since its reconstruction in 1956, the Liederhalle has grown into the hub of cultural and social life in Stuttgart. For more than 50 years now, its distinctive acoustics and architecture have delighted the countless top musicians, artists and orchestras – including international performers – who constantly fill the venue’s halls with music and bring them alive.
Once purely a concert hall, construction of a convention annex in 1991 developed our Liederhalle into an internationally renowned culture and convention centre. This means our abilities extend beyond culture; when it comes to conference business, as a modern event centre we can meet all convention requirements and have the experience necessary to successfully stage conventions, training sessions and seminars.