It's the same in other languages, e.g., in Norwegian, you say, "She had on (herself) a [insert article of clothing].
Hun hadde på seg ei lue.
It means exactly the same thing as, "She was wearing a hat (beanie, toque, woolen hat or whatever)."
I expected this sentence discussion to be like the one in the Norwegian course, overflowing with different names/translations for the Norwegian word (lue) for woolen winter hat!
Literally the the Russian sentence is "on her was a large hat". "На" corresponds to "on".
Russian word for "to wear" is "носить" (it also means "to carry"). It is one of the verbs of motion, and verbs of motion in Russian come in pairs: the unidirectional and the multidirectional verb. You can read about it here
"Носить" is a multidirectional verb when it mean "to carry" and when it means "to wear" it describes a habitual action. "Она носила большую шапку" would mean "she used to wear a large hat"
The unidirectional version of "to carry" is "нести" but it doesn't have the meaning of "to wear", therefore you can't use it for a continuous action "is/was wearing". Instead we use the structure Duo provides here. It's just a quirk of a language.
Unless I'm wrong "шапка" is the feminine nominative of "hat". I believe the accusative singular should be used, so "шапку". Unless it is the genitive singular after "На", but only masculine and neutral nouns can have "-a". Could you please tell me more about the declinaison used here?