When Fritzing was created, we didn't have anything like OSHPark, and most hobbyist were making their PCBs manually at home. That included drilling. That's probably why the Fritzing defaults to 1mm hole sizes with a preety wide copper circle around them — it's simply much easier to do by hand.
But today, when we send our PCBs to fabricators who use automated processes for everything, it doesn't make much sense, and the huge holes and pads mean it's impossible to route a track between them.
Fortunately, you can change the sizes of holes for your headers, using the "hole size" settings:
You have three pre-sets, but you can also enter your own numbers, and they will be remembered as one more preset. Personally I like using 0.9mm for the hole diameter and 0.4mm for the ring thickness — this gives pretty good fit for the male headers, with enough copper to solder them to, and enough room between the pins to fit a trace comfortably.