。
She looked like she was about to faint.
Literal
She [topic-は] faint [looks like-しそうに] appeared.
失神しそうに見えました — she appeared likely to faint. しそうに (looking about to) plus 見えました (appeared, polite past) layers an observer's perception onto the prediction. 失神する (しっしんする) is the medical/formal word for fainting — 失 (lose) + 神 (spirit/consciousness). The polite ました suggests this is being reported to someone.