。
She shook the rug to shake out the dust.
Literal
She [topic-は] dust [object-を] shake-off [in-order-to-ために] rug [object-を] shook.
A clean ~ために ('in order to') purpose construction. The first half states the goal — 払い落とす ('shake/brush off') the dust — and the second states the action that achieves it. 払い落とす is a transparent compound: 払う ('brush, sweep, pay') + 落とす ('drop'), giving 'brush down, shake off.' ゴミ in katakana is conventional for trash/dirt/dust in modern usage; the kanji 塵 or 芥 are older, more literary. 敷物 ('floor covering, rug, mat') is a generic term covering rugs, mats, and tatami — anything spread out on the floor.