。
She held out her hand to shake.
Literal
She [topic-は] handshake [object-を] do-let's [intending-to-と] hand [object-を] held-out.
握手をしようと is the volitional + と + main-verb pattern expressing 'with the intention of, in order to' — 握手をしよう ('let's do a handshake') quoted by と as the purpose of 手を差し出す ('extend a hand'). 握手 itself is a relatively recent vocabulary item in Japan, since the handshake greeting was historically less common than the bow; English-style handshakes are now standard in business and international contexts but still feel slightly imported. 差し出す compounds 差す ('hold out') + 出す ('put out').