。
She was eager to see her family.
Literal
She [topic-は] eagerly family [with-と] want-to-meet.
しきりに ('eagerly, repeatedly') intensifies the desire, and 会いたがった (the past tense of ~たがる on 会う) marks 'visibly wanted to meet.' 家族と会う uses と as the comitative ('with') — meeting WITH someone is a reciprocal action, hence と rather than に. The ~たがる construction stays third-person; if the speaker themselves were the one wanting to meet, they'd just say 会いたかった ('I wanted to meet'). A small grammatical detail with social weight: Japanese is careful about who can claim direct knowledge of an inner state.