If you are adding a graphic to a map doc, it sounds like the box you drew had a fill and maybe covering your labels?? Kinda need more info to be able to help. What environment are your working in, ArcMap, OnLine, etc.
I just checked: No Fill, so there should not be anything Masking the Street Names. I'm in ArcMap 10.3. I'd like to get the street names back and cut everything around the box out. I'm going to start playing with Label Placement options. And Thanks!