。
She has a happy expression on her face.
Literal
She [topic-は] happy-looking [attributive-な] face [object-を] is-doing.
Two patterns combine. The evidential ~そう attached to 嬉しい ('happy') gives 嬉しそう ('looking happy, seeming happy'), the standard way to describe how someone else's emotion appears from outside. Then ~をしている in its descriptive sense — 'has a [feature]' — frames the happy expression as a possession of her face. The pattern X-な顔をしている is the canonical Japanese way to say 'has an X expression': 困った顔をしている ('has a troubled look'), 真剣な顔をしている ('has a serious face'). The polite ~ています form keeps the register neutral.