。
She believes in her own good looks.
Literal
She [topic-は] her own good-looks [object-を] is-believing.
美貌 ('beauty, good looks') is a literary/written word for facial beauty — 美 ('beautiful') + 貌 ('countenance, face'). 信じる ('to believe') with を marks the thing believed in or believed to be true. With ~ている, 信じている is the resultative state of believing — she has settled into a confident, fixed conviction about her own appearance, not a one-time judgment.