The strap goes over the helmet, but the goggles still fit tightly to the face. Racers generally wear the strap outside so the goggles can be pulled down if they get oiled. Motorcyclists often wear the strap outside so the goggles can be raised to prevent fogging when stopped. The only real headache with that is that the straps can slip easily on the helmet, letting the goggles fall off when pulled up. Helmets used to have a goggles strap loop in the back, to prevent that, and some manufacturers offered accessory goggles strap loops, but those have gone the way of the dodo.
P.S. I wear mine outside the helmet. You can't retrain an old racer.