I have been wondering what exactly is a "Unit Stills Photographer" and what is his purpose on the set. What is required to become a Unit Stills Photographer? What qualities and skills do you need to have for this? And what does the Unit Stills Photographer do on the set?

You need to know how to take photos to become a Unit Stills Photographer. 
The Unit Stills Photographer is the photographer that takes pictures with the crew. The pictures are used for marketing purpose.