Many are from the times when the vineyards and orchards got close the city and could only reached on foot.
Slowly the city grew larger and the steps became part of it.
You will find a description (only in German and Swabian - sorry) on this special website
The Tourist Information Stuttgart Marketing organizes regular steps tours for Groups.
Here you can find other individual tours as well as here
And of course the Name "Stäffelesrutscher" is a Kind of nick name for people from a certain region or city and has been used many times here, for example for a theater which presents plays in Swabian dialect.
Here is a short Video of the Stäffelesrutscher-steamtrain:
Very good blog post (in German) with lots of pictures, have a look at this one, too.