If your planning to travel to Leipzig, personally, I would like to give you some advice.
First, the best pages to book your hosting in the city are: AirBnb, Wunderflats, Stayforlong and WG-Gesucht.
About the ideal places to lodge yourself, I would recommend the South West zones such as Reudnitz, or Zentrum Sud. Though, we stayed in the surroundings of Neudstat. The reason behind this recommendation is due these zones are full of. The city center without any doubts would be a great option, but since it is not as economical as the zones we have previously mentioned, it might not be the best idea for an Erasmus+ student.
I shared the hosting with my three classmates in a flat located at Stannebeinplatz. We booked it at AirBnb for a total of 2,400 euros. The host was really kind and caring. Finally, the flat counts with three rooms, balcony, and also has plenty of appliacnes such as: laundry machine, dish washer, blow dryer and even fans for the summer hot weather. Something that I would like to point out is that the flat is nearby a supermarket, a 24h, some restaurants and a tram stop which makes travelling around the city easier.