Being an exchange student at HdM, you won't have to pay any course fees to the Hochschule, though you will be required to pay a contribution to the Studierendenwerk (Student Union) of about 90 EUR per semester.
In our experience, living the life of a student in Stuttgart will probably end up costing you 800 EUR a month. This is the collective cost including accommodation (ca. 290/310 EUR), health insurance ( Non-EU students ca. 90 EUR), food, clothing, mobile phone, culture/tourism, hobbies (ca. 310 EUR) and the costs for public transport (student ticket for the whole semester: 200 EUR).
Add to that the cost of joining some excursions, as you'll surely want to see the sights of the nearby cites!