。
She handed me a postcard.
Literal
She [topic-は] one-counter-for-flat [genitive-の] postcard [object-を] me [to-に] hand-passed.
手渡す is a compound verb: 手 ('hand') + 渡す ('pass over'), giving 'hand over by hand' — emphasizing the personal, direct nature of the transfer. The に marks the recipient (me). 枚 is the counter for thin flat objects: postcards, paper, plates, tickets, blankets. The pattern 一枚のはがき ('one postcard') with の specifying the noun is elaborate but more vivid than the simpler はがきを一枚 — it foregrounds the singular nature of the postcard, suggesting it might matter that it was just one. はがき (lit. 'leaf-of-writing') is the standard postcard, used heavily for New Year's greetings (年賀状) and seasonal cards.