What does it mean to break in a tutu?

Breaking in (much like pointe shoes or jeans) means allowing all fabrics/linings to soften, and stretch out from tension, body heat, and sweat. This process in turn will allow the garment to mold itself to the body shape. 

This is mostly needed for the bodice, basque of a traditional structured tutu.

This is part of the fitting process for professional, pre-pros, and serious dancers.