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She turned her face away so he wouldn't see her tears.
Literal
She [topic-は] him [by-に] tears [object-を] not-be-seen [in-order-not-to-ように] face [object-を] turned-away.
A nuanced grammar showcase. 見られる is the passive of 見る ('see') — '(his) seeing of her tears' = 'her tears being seen by him.' に on 彼 marks the agent (the one seeing). The passive ~ない then negates: 'not be seen.' ~ないように ('so as not to') frames the purpose. Result: 'so that her tears wouldn't be seen by him, she turned her face away.' そむける ('turn away') is the transitive verb for actively turning the face/eyes/body away.