。
She smiled warmly to express her gratitude.
Literal
She [topic-は] warmly smile-and gratitude [genitive-の] feeling [object-を] expressed.
感謝の気持ちを表す ('express the feeling of gratitude') is a slightly elevated, almost speechwriter-style phrasing for thanking someone — softer and more formal than 感謝する ('thank') as a simple verb. 表す ('show, express, represent') here is the standard verb for outwardly demonstrating an inner state. The te-form chain にっこり笑って sets the smile as the manner or accompaniment of the expression — a single composite act of grateful smiling, rather than two separate events.