。
She put on a hat for going out.
Literal
She [topic-は] go-out [for-ための] hat [object-を] put-on.
外出するための帽子 = 'a hat for going out' — the kind of hat she'd wear specifically when leaving the house, as opposed to one she keeps around indoors. The pattern V(plain) + ための + N attaches a purpose to a noun, and is the noun-modifying counterpart of the more familiar ~ために (which modifies a verb). Note that the wearing verb here is かぶる, the one Japanese reserves for headwear: hats, helmets, hoods. Japanese picks a wearing verb based on body region — 着る for torso garments, 履く for leg- or footwear, つける for accessories — so the verb itself tells you what kind of garment is involved. The polite ~ました ending keeps the register neutral and unmarked.