。
She used a spoon to stir the soup.
Literal
She [topic-は] spoon [object-を] use [then-て] soup [object-を] stirred.
~を使って ('using X') is the te-form of 使う ('use') functioning as an adverbial 'by means of' phrase. Roughly equivalent to で (the instrumental particle) but more explicit and verbal — it foregrounds the act of selecting and employing the tool. かき混ぜる is a compound verb: かく (an old form meaning 'scratch / scoop') + 混ぜる ('mix') = 'stir, agitate'. The かき prefix is productive in vigorous-action compounds: かき乱す ('disturb / throw into chaos'), かき消す ('drown out'). Both スプーン and スープ are mid-Meiji loanwords arriving with Western dining culture.