。
She's not the kind of person who gets shy in public.
Literal
She [topic-は] in-public [at-で] shows-embarrassment [like-ような] person is-not.
Three grammar points layer together. The suffix ~がる (from 恥ずかしい + がる → 恥ずかしがる) turns the emotion adjective into a verb meaning 'to show signs of being embarrassed' — required because Japanese restricts direct emotional assertions to first person. The ~ような turns this verbal clause into a noun modifier: 'the kind of [person] who shows embarrassment.' Finally, ではありません is the polite negative copula, completing the negation: she is not that kind of person.